PSA
For the confused who weren't sure what they meant, the terms used in the poll, shounen, shoujo, seinen, and josei are demographics (not genres), much like YA is for novels.
Shounen is aimed at boys, shoujo at girls, seinen at young men, and josei at young women.
They each tend to have recognizable differences in art style, but all span multiple genres.
Shounen is the most widely known, consisting of mostly action, including the "big three" works of Naruto, Bleach, and One Piece. Many harem comedies are also shounen. The most influential shounen work is probably Neon Genesis Evangelion, which ultimately became a cultural phenomenon beyond its target demographic, and is one of the most influential anime of all time, if not the most. Hideaki Anno deconstructs many of the popular shounen tropes, and Eva probably alluded to by other anime even more than Gundam.
Shoujo is often associated with romance, but there are also non-romantic shoujo anime and manga. Its pivotal work is probably Revolutionary Girl Utena, directed by Kunihiko Ikuhara, who also worked on Sailor Moon. Heavily inspired by Eva, it also attempts to deconstruct many of the tropes popular in shoujo manga.
Seinen is what a lot of people point to when they talk about "mature" anime. It includes anime like Cowboy Bebop and Samurai Champloo. What catches a lot of people off-guard is lots of slice-of-life shows like K-On! and Azumanga Daioh are also seinen. If it's a show about cute girls doing cute things, it's probably aimed at young men. A quintessential of seinen anime is Serial Experiments Lain with character designs by the great Yoshitoshi ABe, which is to seinen anime what Evangelion is to shounen.
Josei is pretty much a wildcard in terms of genres, but they do tend to be more contemporary than seinen shows. Honey and Clover is probably the most well-known example of Josei. It also includes works like Nodame Cantabile. Many josei shows like the two just mentioned are set in college or university, but some like Usagi Drop and Space Brothers are about middle-aged characters and beyond.