Why the Wind Blows
Ultimately, the wind blows for the same reasons that the mantle convects and that boiling water in a pot full of spaghetti rises over the heating element of a stove. The air, the spaghetti water, and the mantle are capable of flowing, and all are heated from below and cooled from above. The amount of heating and the rate of flow are VERY different in these different cases, which helps make the world interesting. But you might see a thunderstorm and imagine a hot-spot formation in the Earth, or a pasta dinner, and there is at least a little similarity between a cold front, a subduction zone, and spaghetti noodles sinking along the wall of the pot.
Some of the sunshine that reaches the top of Earth’s atmosphere is reflected from clouds or snow or other things without heating us, but most comes through the air and heats the surface of the Earth, which then heats the air above. The equator receives more sunshine than the poles because of simple geometry. Imagine for a moment that Dr. Alley’s head is the Earth, with his nose on the equator and the North Pole in his bald spot on top. (See the picture.) If he stands in front of a sun-lamp “sun,” he’ll never get a sunburn on his North-Pole bald spot, but he will on his equatorial nose.

The Earth works the same way—at the top of the atmosphere, the amount of sunlight passing through a square meter is the same at the equator as at the pole (see the diagram below). But because of the Earth’s curvature, the light passing through a top-of-the-atmosphere square meter at the equator illuminates a square meter at the surface, whereas the light passing through a top-of-the-atmosphere square meter near the poles is spread over many square meters on the surface, so a square meter on the Earth’s surface gets more sunshine near the equator than near the poles. (Additionally, in both cases, the rotation of the Earth causes the sunlight to be spread around the whole planet.)
The additional energy received at the equator compared to the poles means that the surface at the equator becomes hotter than at the poles. If we had no atmosphere or oceans, the equator would become too hot for life as we know it, and the poles too cold. However, the extra sunlight that the equator receives heats air and water there, driving winds and ocean currents that carry some of the excess heat toward the poles, making the whole world habitable to humans.
Video: The Drivers of Earth's Weather (1:34)
Dr. Richard B. Alley: We're going to look here at the main driver of Earth's weather. The sun is this big orange thing over here. It is sending energy towards us - beaming along like this. You have a spaceship. Your spaceship has solar cells on it. It doesn't matter whether your spaceship is sitting over the equator or over the pole. You get about the same amount of energy in either case. But when you come down to the surface, if you add your solar cell facing straight up to space at the equator, you'd get almost as much energy, unless it was cloudy, but when you put the same size solar cell facing up to space up here, it doesn't get as much energy. There's more coming through here that's being spread over a larger area and because of that, the heating is greater at the equator than it is at the pole. The greater heating causes the air to rise here. It heads towards the poles, like this. It blows over the ocean and that can make currents. The air, the winds, and the currents are moving the extra heat from the equator to the pole. Eventually, it's all radiated back out to space so we're close to balance, the Earth rotates so there's a little complexity there but the big picture -- because the equator faces the sun and the pole doesn't there is more heating at the equator.
Following is a static image that was described in the video above.

In slightly more detail, as the sun heats the land at the equator, the land heats the air above, and the air expands, rises, and then moves poleward in convection currents. The hot air loses some of this energy to warm the land and water it passes along the way, but eventually, all the energy is radiated back to space. The sunlight that comes in is called shortwave radiation because its waves are short. (This stuff really isn’t that complex a lot of the time!). The radiation going out has longer waves and is called longwave radiation. Your eyes can see the shortwave light, but you would need infrared goggles to see the longwave. Later, we will discuss how the different way that shortwave and longwave radiation interact with certain gases in the air is important in understanding the greenhouse effect. For now, note that the global energy budget is very close to being balanced—the total amount of energy brought in by shortwave radiation and absorbed in the Earth system is very nearly equal to the total amount of energy taken out by longwave to space. (This is not perfectly balanced now, because we humans are changing the composition of the atmosphere by adding greenhouse gases, so the Earth is warming because we are sending out a little bit less energy than we receive. But, once we quit changing the composition of the atmosphere, the Earth will get back to balance.) A factory balances the total amount of stuff coming in and going out, but little auto parts come in and big cars go out; the earth balances the total energy coming in and going out, but shortwave comes in and longwave goes out. And, the uneven heating because of the Earth’s nearly spherical shape drives the wind, and the wind plus uneven heating of the oceans drives ocean currents.
Because the Earth rotates, the winds end up turning rather than going straight from the equator to the pole, and this makes the weather much more interesting than it would be on a non-rotating planet. If you want to explore this a little more, see the optional Enrichment about the Coriolis effect.