BGM: Background Music. This one is pretty self-explanatory. It's music that is played during the background of an anime, usually instrumental. Vocal songs played during the background of an anime can also be called 'Insert Songs'.
Bishoujo: Pretty or beautiful girl.
Bishounen: Pretty or handsome boy. These are those gorgeous, vaguely (or very) effeminate anime guys who are usually so 'pretty' they can be mistaken for girls. They can sometimes be characterized by slender body build and long flowing hair of an unnatural color. Bishounen often 'sparkle', be it shimmer sparkles that hover in the air around them, or a sparkling lens flare that reflects off their shiny white teeth. Lately 'bishounen' has also been used to describe any anime guy a fangirl thinks is really cute. For example, Hotohori and Nuriko in Fushigi Yuugi are bishounen.
Doujinshi or Doujin: A manga that is written/drawn by a fan or an amateur. Doujinshi can be a fan created original work, or can use existing characters. There are many types of doujinshi -- dramatic ones that place popular characters in new adventures, parody doujinshi that have lots of gags, adult doujinshi (which seems to be the most common type that makes its way over to the US), and others. Doujinshi are so popular that there is a huge doujinshi-selling convention every year in Tokyo called 'Comiket' or 'Komike' (Comic Market) that usually brings in around 300,000 attendees!
Chara: Shortened version of the word 'character.'
Circle: A club or group that is formed to draw or create doujinshi.
Cosplay: The Japanese version of the term 'Costume Play'. It is the act of dressing up in costume (usually as an anime or videogame character) and possibly performing a skit related to that character. Cosplay can also refer to the actual organized event where people act out their skits.
Ecchi: A mild version of the word 'hentai.' Ecchi is generally used in a similar way as 'naughty', meaning 'sexual naughtiness.'
ED: Ending song. The song that plays at the end of an anime, usually with the ending credits.
Eyecatch: The little bit of new animation leading in and out of commercial breaks. Sometimes even OVAs will have an Eyecatch midway through even though it is not broadcast on television. It's a good way to divide the story content.
Fan Service: Something in an anime that is done with the intent to appeal to the male audience. This usually ends up being gratuitous shots of female anime characters. I suppose there can be fan service for females, I'd say a lot of the second half of Shoujo Kakumei Utena had female fan service. But the male fan service moment that pops to mind is the Minmei 'shower' scene in Macross: Do you Remember Love?
Hentai: Japanese word for pervert. This strong word is generally used for more sexually vulgar/perverted things or people.
Insert Songs: Vocal songs used in the background of an anime.
Image songs: Songs that are not necessarily in the anime (and are usually not) but are related to the anime in that they are sung by an anime character's voice actor while in character or are inspired by the series.
Kawaii: Japanese word for cute or adorable. This word is usually used for younger girls. Girls themselves use this word a lot in regards to everything!
Manga: Japanese comics. In Japan 'manga' can sometimes refer to anime too, but for the most part it's comics. Japan is a huge place for comics. Everyone reads them, so they publish stuff marketed to everyone from 1st grade girls to working businessmen. Manga usually runs in periodical magazines (like weekly, biweekly, or monthly magazines), and the story installments are gathered together and published in volumes.
Mecha: Word often used to refer to any large mechanical thing such as a giant robot.
Omake: Means 'extra' or 'bonus'. It's like a short comedy skit at the end of an anime. Examples of series with Omake would be Blue Seed, Video Girl Ai, and the Fushigi Yuugi OVAs.
OP: Opening Song, the opening song for an anime.
OST: Original Soundtrack. Yep, this is the CD soundtrack for an anime. They usually contain the BGM used in the anime as well as some Insert songs. There are also some 'Image CDs' which contain music not used in the actual anime, but instead has music 'inspired' by the anime, or songs sung by the voice actors in character. There are often several OSTs for an anime series.
Otaku: Basically an obsessed fan, and not necessarily of anime -- for example, you can be a trading card otaku or a Star Trek otaku. The word has some negative connotations in Japan. It usually brings up the image of a smelly, socially challenged obsessed fan, but in America, anime fans have taken the word and made it their own and usually use it with pride to mean anyone who is a big fan of anime, without the negative meaning. So while some fans go by the Japanese way of associating the word with a negative image, others embrace the word and use it with the American meaning.
OVA/OAV: Original Video Animation/Original Animation Video (they both mean the same thing and are used interchangeably). This is basically any anime that is a direct-to-video release. It has not been shown in movie theaters or on TV. Several OAVs can make up an OAV series, with each part being released on a separate tape or DVD. OAV series are often recognized as having better animation than most TV series, and shorter storylines. Examples of OVA series that are being released in America are the Video Girl Ai series and Dragon Half.
Seiyuu: A Japanese voice actor or actress. In Japan, seiyuu are used to dub all sorts of things, not just anime. So there can be seiyuu who do voiceovers for Western TV shows or movies too. In Japan, there are schools dedicated to just voice acting!
Shoujo: Technically means 'girl'. Usually used to refer to manga that is targeted towards girls, or to anime that is based on a manga story that ran in a magazine targeted towards girls. Characterized by the focus on relationships/plot over action, but there are always exceptions.
Shounen: Technically means 'young boy' and is usually used to refer to anime or manga targeted towards boys. Characterized by lots of action and adventure instead of relationships or romance.
Super-Deformed (SD): Describes when an anime character suddenly changes from looking normal into a smaller version of themselves, with less physical detail, shorter limbs, larger head, and larger eyes. Characters often turn SD as a humorous effect, or in response to an outrageous situation. This happens a lot in Magic Knight Rayearth.
Tankoubon: A volume of manga that consists of all the serialized manga story chapters that ran in a magazine gathered together and printed as easily collectible books. Can be similar to an American 'Graphic novel.