The only way that the Earth is heated, clouds form, snow falls, cars run, and people exist is by energy being spent or transferred. The universe, depending upon the theory of the day, has a finite, yet inexorably large amount of energy. The amount of energy available for anything hasn't increased, and is relatively constant across the solar system, with only excruciatingly minute depreciations over time.
Energy exists in many forms, but by definition it is "The ability or capacity to do work on some form of matter". An object can be kicked, heated, cooled, lifted, etc., and all of that is done through energy. Likewise the results of all those actions (a kicked object going through a window, a heated thing burning your hand, a cooled body getting hypothermia, a thrice-lifted couch causing much complaining, etc) is done through energy as well. Often the amount of total energy available for an action is relatively the same. For example there is no difference in the amount of energy necessary to toss a baseball up in the air to a certain height, and then have it drop back down to the ground. "Internal" energy is the total amount of energy stored in an object, whether it be an individual atom or an entire thunderstorm. This internal energy is also an indication of the amount of work an object is capable of doing. Since an atom itself possesses the greatest amount of energy, it's nearly impossible that an object use a significant part of it's energy to do something. For our purposes, internal energy is comprised of three basic forms: Potential Energy, Kinetic Energy, and Heat.
represents obviously the potential for an object to do work. An easy example of this would be the bucket placed over a door filled with all sorts of nasty waiting for a victim. The main source of this energy is through gravity. The amount of gravitational force that can be used on an object, though it hasn't fallen yet, comprises the majority of tangible potential energy examples in the world. In meteorology, potential energy is by far the largest energy source. All the water in the atmosphere and oceans exists mostly in potential energy, not necessarily because of gravity, but because of its propensity to be lifted, dropped, heated, and well...because there's just so much of it.
Kinetic Energy is most often seen in the form of movement, in fact, any moving object, molecule, or atom possesses and exhibits kinetic energy. The faster something moves, the more kinetic energy. Weight is also a factor here, as a moving truck has more kinetic energy than the same volume of air in a gust of wind. As was mentioned earlier, the way temperature is measured is through the average speed of molecules, which is kinetic energy. All matter has kinetic energy because the atoms within it are moving within it.
The First Law of Thermodynamics states that "Energy cannot be created, nor can it be destroyed." It is also known as the "Law of Conservation of Energy". So basically we're looking at energy just being transferred from one object to another, and from one form of energy to another. In weather, this is most often experienced as Heat.
Heat is a subset of kinetic energy, and represents energy in the process of being transferred between two objects (or between parts of the same) because of their temperature difference. When something is being cooled, think of it more in terms of heat being removed. There is no such thing as "putting cold into something". After this heat is transferred, it now becomes part of the internal energy of the new object. If you boil water, heat is being transferred from a burner into the water. Even after the heat source is removed, you can clearly see the effects of the increased energy within the newly heated water. We don't have any large burners in the atmosphere, just as we don't have any airborne pots of water, so how is heat transferred in the atmosphere. It is achieved through Radiation, Conduction, and Convection.
Water exists in the atmosphere primarily as water vapor, a gaseous state of water molecules. An element or molecule can exist an any of three states: Gas, Liquid, or Solid. Water vapor is the gas state of what is otherwise seen as water itself, and ice. In order to go back and forth from one stage to another, a certain amount of energy is required. If you heat water enough, it changes to water vapor. We can see a transformation like this every time we step out of a swimming pool. Water on your body is evaporating as water molecules are escaping to join those in water vapor form in the atmosphere. 
The faster molecules obviously escape first, and since temperature is measured by the average speed of the molecules in question, those water molecules left on your body will have a cooling effect on your skin. In situations where a molecule changes from a liquid form to a gas form, but doesn't change any of the external qualities of itself (temperature, pressure, etc.), it is said to acquire heat energy called Latent Heat. It is heat energy that may be sensed on one side (the cooling of the body or the use of a burner) but not on the other (water changing to steam but with no apparent temperature change). The energy is still transferred from the liquid form of water to the gas, but does not show itself in temperature. The reverse process, which would then exhibit that heat energy (ice melting, etc.) is said to show Sensible Heat. It is deduced, therefore, that evaporation is a cooling process, and condensation is a warming process. Whenever heat energy is being absorbed to make a phase change in a substance (for our purposes water), that heat is "latent". When it is released in the reverse process, it is called "sensible". If you boil water, heat is being transferred from a burner into the water. Even after the heat source is removed, you can clearly see the effects of the increased energy within the newly heated water. We don't have any large burners in the atmosphere, just as we don't have any airborne pots of water, so how is heat transferred in the atmosphere? It is achieved through Conduction, Convection, and Radiation.
Conduction is the transfer of heat from one molecule to another within the same substance. For example, the reason why pouring a hot cup of water into a cold bucket of water doesn't create a small area of water that's hot surrounded by cold water is through conduction. Another way you can show this is by putting a piece of ice over a clean metal bar and one over wood. The clean metal bar can 'conduct' heat (or electricity for that matter) much better than a block of wood. You will see the ice melt much faster on the metal bar as the the temperature difference between the ice and metal is dispersed rapidly and evenly across the metal. In our atmosphere, air is quite terrible at conducting heat. It is therefore said to be a good "insulator", and is often used between two surfaces for insulation against rapid temperature changes. Because air is such a poor conductor, the ground when heated by the sun would only warm a layer of air right near the surface an inch or two thick, yet we know that massive amounts of heat are moved and expressed in the atmosphere all the time.
Convection is heat transfer by means of a mass movement of a gas or liquid (such as water vapor or just plain water). If it's possible to set up a current within it, it can convect. In our atmosphere, the heated air right at the surface (that measly few centimeters caused by conduction) begins to expand as it is being warmed by the ground. That expanded air becomes less dense than it's surroundings, and as mentioned earlier, it starts to rise. This allows large amounts of heat energy to be transferred upward in the atmosphere. Conversely, cooler air would sink lower, and replace that departing warm air. The cooler air would then be heated by the Earth's surface, and rise. This is known as convective circulation. Though convection actually signifies any transfer of heat through these means, meteorologists use the word only in the sense of vertical air motion, and they use the word Advection when referring to heat transfer by horizontal winds.
Radiation is the other mechanism for heat transfer in the atmosphere. This method allows heat to be transferred from one object to another, without the space between being heated. This can easily be experienced by sitting around a campfire. You feel nice and warm, even though the air still has quite a chill to it. This radiant heat energy travels in waves and doesn't need molecules to transport it. Once this energy is absorbed by an object, energy is released, resulting in some sensible heat. All things that have a temperature above absolute zero (Read: ALL THINGS), radiate energy. This energy is emitted in waves, whereby higher temperatures emit more radiant energy in shorter wavelengths. It is difficult at this point to discuss thermodynamics credibly without throwing a few pages of advanced math up, but suffice it to say you can see this at work when an object is heated enough so that it glows red. This basically shows that radiant energy is being emitted at a wavelength that the eye can discern, and does so as red light. In our solar system, the sun is the majority source of radiation, and radiant heat energy because it has by far the highest temperature (around 10,500 F), but the Earth and atmosphere can also play a role here.
If all objects only radiated heat energy, everything would just get progressively colder, but it's a good thing that all objects absorb energy as well. If something absorbs more energy than it radiates, it gets warmer. Likewise, the reverse is true. If an object radiates the same amount of energy as it absorbs, the temperature will remain constant. The Earth and Sun are pretty close examples of what is known as "Blackbodies", objects that absorb all the radiation that strikes it and emits the maximum radiation at a given temperature. Our Earth sits half in sunlight and half in darkness at any given time of day. The side of the Earth in sunlight is absorbing solar radiation while the side in darkness is emitting radiation. At some point, every interaction between two radiating objects strikes up a balance. This is called a "Radiative Equilibrium Temperature". The Earth, being a shade under 100 million miles away from the Sun, has a radiative equilibrium temperature of about zero. That's right, the amount of solar radiation making it to Earth gives us enough heat to be zero degrees. But why is it warmer than that? It's time to step aside and thank 'greenhouse' gases for all the hard work they do...
The Atmosphere, Radiation, Absorption, and the Glorious Greenhouse Gases
The atmosphere is also able to absorb and emit radiation, and likewise it has a temperature, but it is not constant, and it is also an imperfect radiator and does not behave like a black body. Different elements in the atmosphere absorb certain wavelengths of energy, and they also emit the same wavelengths of radiation, although if you move across in wavelengths, any certain molecule may be particularly poor at absorption, but extremely good at others. For example, Ozone is very good at absorbing UV rays (very small wavelengths, and thus quite powerful radiative energy), which is good because we really don't want to. Methane, Nitrogen, Carbon Dioxide, Water Vapor, and all other atmospheric compounds each absorb certain specific wavelengths of radiation, and can also radiate that energy back to space or towards the Earth's surface. The best example of this is Carbon Dioxide. CO2 just so happens to be quite a good absorber of the Earth's radiation (remember, because the Earth has a temperature, it also radiates heat energy). CO2 Is also present in large quantities in the lower atmosphere. When it absorbs this energy, carbon dioxide radiates this energy in all directions (as all radiating bodies do). The Earth would then get some of that heat energy back, reradiate it out, and thus would form a cycle of radiative transfer that would establish an equilibrium temperature at a much more livable 59ºF (keep in mind that Carbon Dioxide is just one of the gases that helps achieve this, though it is the main gas).
The term "Greenhouse Effect" has gotten a lot of bad press in the last, well, since it has existed in terminology, but it obviously provides an absolutely vital process to our atmosphere. The concept of a greenhouse isn't even good for describing what happens in the atmosphere, because the warmer air inside a greenhouse is due primarily to the inability of the air inside the greenhouse to mix or circulate with air outside of it. Though it's too late to change anything, we should probably think of a better name for it, one that simply denotes the atmosphere's ability to heat the Earth to a temperature much higher than it otherwise would be.
What about incoming radiation that is not absorbed? Well, much of it is reflected. Reflected radiation depends on an object's Albedo. Albedo refers to the percentage of radiation that is immediately reflected outward. Reflected radiation is not the same as emitted radiation, because...well, let's just give the example of snow. Snow reflects about 95% of all solar radiation that strikes it. Snow is quite cold. Therefore, the majority of radiative energy one is receiving while skiing comes from the sun, both directly to the body and that which is reflected by the snow. The snow itself contributes very little, because of its cold temperature. It is no small worry then, large painful sunburns on those ski slopes. Water is a very weak absorber, where only about 10% of incoming radiation is reflected. The Earth as a whole, has an albedo of about 30%, meaning of all the insolation it receives, 30% goes right back out without even leaving a mark.
Putting this all together, the Earth has quite a complicated heat-energy-balance system. It is much more than just the Earth and the Sun fighting it out for a final temperature of 0ºF. We know pretty much the equilibrium temperature of the Earth is about 59ºF, so let's examine how that is concluded with a look at the "Earth's annual energy balance", or "heat budget".
The Earth-Atmosphere-Solar Heat Energy Balance
This graphic shows what happens to the approximately 342 Watts per square meter of solar energy that enters the Earth's atmosphere. You can think about in terms of percents and balance. About 30% of the insolation is reflected right back out, or 107 Watts per square meter. The Earth's atmosphere also plays a part in radiating part of the Earth's heat energy back to the Earth, and the rest to space, thus ensuring a multi-level balance. The amount of solar radiation in is balanced just about perfectly with the amount going out. Likewise, the atmospheric "greenhouse" gases that re-radiate heat back to the Earth's surface establish a balance with it such that we get that oh-so-nice boost in average surface temperature from 0ºF to about 59ºF. Note that any little change in any one of the properties at work can actually manipulate this balance and establish a new, perhaps temporary equilibrium. Solar sunspots, seasonal increases in carbon dioxide, and even man-made influences can automatically cause the Earth to adjust to higher or lower amounts of heat energy. The resultant effects of these minute perturbations is one of much study and even more consternation of the environmental community. So far, however, the planet Earth has been quite good at being a "Gaial" system, regulating itself almost as a living organism to restore balance throughout its skies and shores. It's always been interesting, however, that massive storms and systems that demonstrate natures supremacy over human existence are created out of a mere need to move some heat over to another side of the map.
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