The westerlies, also known as the prevailing westerlies, are the dominant wind pattern in the temperate latitudes, meaning everything between an area around the tropics to the arctic. Of course, to say something is the dominant wind pattern is not to say much, because the changes brought about by terrain, storm fronts and the infamous butterfly wings can change many other weather patterns. However, all things being equal, the westerlies are the main shaper of the weather for close to half of the planet's population.
The basic circulation of the atmosphere is caused by heating near the equator, which sets up a conveyer belt of sorts where hot air is sent poleward to be cooled, and cold air is sent towards the equator to be heated. This is a basic cycle of convection that can be observed in a frying pan full of water on a stovetop. Other than terrain, the main thing that interferes with this is the rotation of the earth. The Coriolis effect redirects air against the direction of rotation as it heads north. Or rather, it appears to redirect it: the truth is, the northward flowing air continues on its course, the ground underneath it is just rotating at a different speed. In either case, this is why at the middle latitudes, the wind mostly blows from the west.
While the explanation is simple, the effects are gigantic. One of the largest effects, and most important, is that the western side of continents are often more temperate than the eastern sides. On the west side of the continent, the westerlies are coming off of the ocean, which moderates their temperature. On the eastern side of a continent, the westerlies are coming from the interior of the continent, meaning that they are much colder in winter and much hotter in summer. This is the reason that Portland, Oregon has a much more mild climate than Montreal, Quebec; and why Venice, Italy has a much more mild climate than Ulan Bator, Mongolia or Vladivostok, Russia.
On a personal level, the westerlies play an important part for me because first, even close to a thousand miles inland, I still enjoy a somewhat more mild climate because of them. And just as importantly, they provide pattern and form to the world. The sun, moon and stars rise in the east, and almost as reliably, the winds, clouds and weather come from the west, cresting over the mountains. It is the result of some complicated applications of the second law of thermodynamics, on a rotating sphere, but yet looking to the west for the weather is close to instinctive for me.