Shounen means "for boys" and refers to anime marketed at boys. This especially includes action, harem anime and mechas; well-known examples of each of these, respectively, include Dragonball Z, Tenchi Muyo and the Gundam series. Harem anime refers to a show in which a male lead lives with, works near or is otherwise involved with a large cast of desirable female characters and inevitably romantic or mildly sexual comedy ensues. Mecha anime is defined more thoroughly below.
A more grownup form of shounen anime is called seinen, which is marketed at young adult men.
Mecha anime is considered a subgenre to shounen, but is so enormously popular that it's deserving of its own section. A mecha is a giant robot that may operate independently, but more commonly contains a pilot inside who controls the robot like a suit of armor. The aforementioned Gundam series is probably the most well-known, and it's not uncommon for the mechas to be referred to simply as Gundams. Other popular mecha anime include Macross, Evangelion and Tengen Toppa Gurren Lagann.
Shoujo means "for girls." It can include almost anything from romance and drama to action to comedy. Its primary distinguishing factor is its focus on the characters and the relationships between them. There is a shoujo subgenre called reversed harem, which is essentially the same as the harem anime described in the shounen section, but with a female lead surrounded by male characters. Popular shoujo titles include Ouran High School Host Club (a reversed harem anime), Fruits Basket and Nana.
There is a genre aimed at somewhat older women called josei. It mainly includes shounen ai and yaoi. The former is romance between homosexual males, and the latter is much the same, but includes explicit sexual situations. There are a few josei titles unrelated to shounen ai and yaoi, but they're rarely if ever translated into English.
Mahou shoujo, meaning "magical girl," is one of the most prolific subgenres of shoujo. These anime feature a female protagonist comparable with a superhero, who typically obtains some kind of magic power that allows them to transform into monster- or crime-fighters. Examples of mahou shoujo anime include Sailor Moon, Cutie Honey and Princess Tutu.
Hentai refers to animated pornography. Ecchi is derived from the Japanese pronunciation of the letter H, which stands for hentai. However, ecchi has grown into a separate term in its own right and can be used to refer to anime with a great deal of sexual humor or other sexual content that isn't explicit enough to be labeled hentai.