Aesthetics is the branch of philosophy that studies perception and beauty.
It describes how we feel and interpret our experiences, and how we react emotionally to certain stimuli. Among these, the most interesting is the feeling of beauty. Aesthetics provides a basis for the fine arts: painting, drawing, theatre, sculpture, music, etc., providing an understanding of how these things affect us as individuals and as a society. Some philosophers view aesthetics as the least important branch of philosophy because it is so dependent on the other branches — before one can study beauty, one needs a reality in which to observe it, a mind to perceive it, a society to interpret it, and a value system of some sort to judge it. Others prefer it because questions of aesthetics occur more frequently in practical matters. Aestheticians (as opposed to anesthesiologists, who make you feel nothing) tend to be more down-to-earth than, say, philosophers of mind.
From the cradle to the grave, we live in a world of perception. Sight, by far, is the most primary of the six senses, but we also base our daily life on our ability to hear, touch, taste, smell, and balance. One of the first questions to be asked in aesthetics was, "Can we depend on the senses to give us good information?" While things would be much easier if this was the case, most philosophers agree that our sensory equipment can be easily fooled.
This is incredibly easy to demonstrate, thanks to an experiment due to John Locke. Take three dishes of water: one hot, one cold, and one lukewarm. Put one hand into the cold water, and one into the hot. After a few minutes, place both hands into the lukewarm water. Locke found that both hands disagreed with respect to the temperature of the water; the hot one believes it to be cold, and the cold one believes it to be hot. A thermometer, on the other hand, reads it at 17 degrees. Similar faults can be found with the other senses, as well.
The history of aesthetics is marked by an identity crisis. The greek word αισθητικη (aisthetike) from which we get the word "aesthetics" means "sensitive person" or "perceiver". It wasn't until Immanuel Kant used the term to describe the study of beauty (which he stole from Alexander Baumgarten) that aesthetics became known popularly as just that. This part of aesthetics tries to unpack the experience of beauty and determine what laws, if any, beauty obeys. Of course, the German philosophers weren't the first to discuss this strange thing, beauty, but they were the first to name it as aesthetics.
Beauty is a very strange thing, indeed. Two lovers, having seen the same movie (though perhaps not much of it) will often quarrel over whether it was worth the price of admission. There are fierce divides in the taste of music; for example, those who love country are defamed by those who hate it. Nobody seems to agree whether rap or hip-hop is art or dross. Pop music too has come under fire by those who believe it is produced mechanistically by the big labels, drowning out the culturally superior indie music. Then there are art critics who prefer Vincent Van Gogh to Pablo Picasso and the other way around; there are food critics who despise haute cuisine and those who won't eat anything but.
Whether beauty is objective, socially defined, or completely dependent on the individual (that is, in the single eye of the beholder) is not the only fierce debate in aesthetics. What can be beautiful? Pal Erdos believed firmly that mathematics was driven by beauty, and Paul Dirac believed that physics should be guided by beauty in its equations. But the beauty of an equation is not in the appearance of the symbols used to write it, but in the meaning of the symbols. The study of symbols (made popular by that horrid book), semiotics, could be thought of as a branch, or maybe an offshoot, of aesthetics.
Studying aesthetics helps one develop a more refined sense of the beautiful.