Anime is very hard to describe in only a few words. If you don't know what anime is, here are a few words that might help you get started. Anime is Japanese animation. The word anime is a transliteration of the English word animation. Related to anime is manga. Manga is what comics are called in Japan. The word manga was coined by the artist Hokusai in 1815, usually translated to mean "irresponsible pictures."
Anime and manga are significantly different from their counterparts in America. For the most part, cartoons and comic books in America are thought of as geared toward youngsters. Cartoons and comics books in Japan are geared towards all ages. While there is a large amount of American animation and comics which are intended for older, more mature readers without the need for bulging superheroes, they are seen as something separate from the conventional line of comic books (hence DC Comic's Vertigo label as a separate entity from its normal titles). Millions of subway riders in America pass their time reading novels on the train. In Japan, you're just as likely to see someone reading a comic book.
While well thought out, "adult" animation is something of a rarity in America there are pleanty of tv shows, movies and direct to video (called Original Video Animations or OVAs in Japan) examples of anime. Look for high quality animation, real vocal acting (voice actors and actresses are big stars in Japan who hold concerts and have magazines and "data books" devoted to them), music everywhere, real drama like people dying and falling in love (sometimes at the same time!), and even the occasional English phrase thrown in for no apparent reason.
Clearly the above doesn't apply to all anime. There are good shows and movies just with any sort of entertainment. But the only way you'll know for sure how right I am in my description is to watch it for yourself. ;)
Anime on American TV
You might have seen anime on TV and else where! Over the years it has been molded to fit American audiences (or at least what TV producers think American audiences are). If you've lived in...
the 1960's... you might have seen Speed Racer, Kimba the Lion or Astro Boy.
the 1970's to the early 1980's... you might have seen Battle of the Planets and Star Blazers.
the mid-1980's... you might have watched Robotech and Captain Harlock and the Queen of a Thousand Years on TV or played Cliffhanger and Cobra Commandin the arcades.
the 1990's... You might have seen The Ronin Warriors, Sailor Moon, Technoman or a multitude of anime actually being referred to as anime on TV, in stores and in the theaters.
You already know anime and manga from above. Here are some other, re-occuring words and phrases you'll come across while watching anime or talking about it with friends.
Baka! -- "You're stupid!"
chara -- character
chibi -- Short, tiny. A chibi character may naturally resemble a
super-deformed character, where the entire body has a smaller,
squashed, cuter appearance. If you need to get a feel of what
this means, try The Chibi & Super-deformed Image Archive.
dojinshi -- Generally fan stories pertaining to a specific series.
Could also be any series specific fan art.
hentai -- Abnormality. Sexual perversion. Used also to describe
a certain class of anime, manga or otaku (as in "H-Anime").
idol -- (a) Certain people and characters transcend being
something that you're a fan of. When you start worshipping them,
adorning yourself in their image, collecting dolls, images
or cards of the character, they've become your idol.
(b) Singers with a major role in a series (Minmei from
Macross for example).
Japanimation -- American slang for for Japanese animation.
Similar to other less than ideal contractions like "blaxsploitation"
and "teensploitation." It's best to either to use the full
"Japanese animation" or shorter "anime." Don't talk about
"Japanimation" to an otaku.
kawaii -- cute
henshin -- Transformation. Used to describe the sub-genre
of special-effects shows featuring super-hero transformations
(Kamen Rider or Metal Heroes). Also used to describe transforming
mecha (Henshin Robo).
mecha -- Just about anything mechanical: weapons, vehicles,
robots, but usually not items like pencil sharpeners or ceiling fans.
Nani? -- "What?"
Nihon -- Japan.
otaku -- Anime fanboy or fangirl. Generally a good thing in America,
aparently closer to being a geek in Japan. If you're wondering if
you've become an otaku take Axcess magazine's Are You An Otaku? test.
OAV/OVA -- Original Video Animation: an anime direct to video
release, anime made only for the video market. Direct to video
releases are common in Japan, while still a rarity in America.
Unlike the US, where Direct to video movies are still often seen
as a reprieve from what would've been a disasterous theatrical
release, OVA's in Japan can be big events unto themselves.
seiyuu -- Anime voice actor or actress. Hitoshi Doi's Seiyuu
Database is the best place to go for more information.
sentai -- Task-force. Used to describe the task-force genre of
live action TV series, including Zyuranger/MMPowerRangers and
DaiRanger. Some American fans have been known to use this as
a synonym for special-effects shows when they really mean to
shojo -- Type of anime or manga targeted to the "young girl"
shonen -- Type of anime or manga targeted to the "young boy"
tokusatsu -- Special-effects. As one might use it in English,
tokusatsu is used to describe a super-categeory of live action
super-hero shows (Kamen Rider/Masked Rider or Zyuranger/MMPowerRangers)
and giant monster movies (Godzilla or Mothra).