The theme music for most anime tends to be catchy songs deliberately written for release to the pop/rock music market, if they aren't already actual pop/rock songs. One reason is simply that anime provides an easy way for both hit and entry J-pop/J-rock artists to get more exposure and good lateral promotion. Platinum J-rock bands like L'arc-en-Ciel and Orange Range frequently release their new songs along as themes in anime that are on the air at the same period as their respective singles or albums.
Another reason this is done is because many anime voice actors are also singers, often the more successful ones. (At least one such performer, Megumi Hayashibara, is both a formidable presence on Japanese pop charts and an internationally-known talent, as well as the recipient of more star and featured anime roles than any one person ought to have.) It's not unknown for production companies to organize some of their principal cast members into groups for recording CDs — the "Goddess Family Club" (Ah! My Goddess), DoCo (Ranma ½), the Maho-Dou (Ojamajo Doremi), Peach Hips (Sailor Moon) and the Spirit Singers (Digimon Frontier) all come to mind. Either way, it's usually to a voice actor's advantage — they perform theme songs (as well as additional "character" songs), receiving a double benefit from exposure in two different markets (and the additional profit).
A third reason is that TV theme songs are the pinnacle of Japanese musical success. If a Japanese artist/group makes a hit album, the studio takes almost all of the profits. If the same group makes an album as a TV tie-in, the musicians themselves receive a much larger cut.
As a result, many anime theme songs have little to do with the subject matter of their shows. To western ears, it may just seem like they're singing about fate, destiny, and really nothing else in particular. Many are romantic songs of one flavor or another, ostensibly showing the point of view of one of the show's main characters. (This isn't limited to shojo or josei - there are plenty of cases of effeminate love songs being used for shonen anime.) Alternately, they may be Thematic Theme Tunes, reflective "personality" pieces, nonsense patter songs, or instrumentals.
Whatever their style and content, though, anime theme songs are generally written and performed with the same attention and care that in the United States is reserved for potential Oscar-winning compositions. Quite a few can reach Ear Worm status - one example, "Hare Hare Yukai" from Haruhi Suzumiya, has been wildly popular as both a song and a dance at American and Japanese conventions.
When an anime reaches the American and International broadcast market (as opposed to direct DVD sales), their theme songs are often either shortened or changed entirely. When a broadcaster does use new credits (Toonami and other outlets are notorious for not bothering to do so), a vocal performance may replaced with instrumentals; a case in point would be The Vision of Escaflowne, whose vocal song was traded for an "adventure-style music" opening. Other times the original melody is kept (perhaps with a little modification), and new English lyrics unrelated to the original are written for it; an example of this would be Sailor Moon, whose theme, "Moonlight Densetsu", was turned into a standard Western Expository Theme Tune Roll Call.
On the other hand, some importers have tried to create local language versions of theme songs faithful to the original Japanese lyrics, with mixed results. Difficulties have included license constraints on North American distributors from Japanese parent corporations, and the problem that American voice actors are rarely trained singers - as Viz's famously bad attempt to create "DoCo America" proved. In the late 1970s, Space Battleship Yamato's theme was dubbed reasonably well, even allowing for the changes that turned the series into Star Blazers, and the same was done in the 1980s with some of the themes from Ranma ½. Pioneer (later known as Geneon) sometimes did the same in following decades, even going so far as to release full English-language CDs for some of their imports. Funimation has revived the practice, with mixed results. Mexican and Chilean dubs of anime have also translated some songs, often with good results; the Latin Spanish versions of Mazinger Z, Science Ninja Team Gatchaman, Dragon Ball Z and You're Under Arrest! themes are as memorable and loved as the originals.
Very rarely, a replacement theme will prove to be more appropriate for English audiences (the Toonami run of Mobile Suit Gundam Wing's replacement of Just Love with an instrumental version of the show's original op being the most famous example). When this is successful, it's generally because the new theme stays more in step with English viewers' preconceptions of a series' tone.
One other thing of note concerning theme songs for anime: where in the United States a theme song is usually a vital part of the identity of a show, anime often change both opening and closing theme songs on a regular basis. The best example of this would be (again) Ranma ½, which had a different set of theme songs for each of its seven seasons and for its OVA series. Another good example is One Piece, which has (thus far) 18 openings and 18 endings in 692 episodes, or Detective Conan, which so far has a whopping 39 openings and 49 endings. Similarly, episodes may feature several different renditions or versions of the same theme, as with "Fly Me to the Moon" from Neon Genesis Evangelion or "A-LY-YA!" from Negima!?. z The original lyrics to an anime theme song may be the occasion for Gratuitous English — the number of examples where English words and phrases are used instead of Japanese equivalents is vast. "Treat Or Goblins", the theme from Magical Shopping Arcade Abenobashi, contrasts Megumi Hayashibara's attempt at half-English hiphop with an all-English rap by an obviously American performer. In a few cases, the song is actually written mostly or entirely in English and performed at least in part by native English speakers — "Duvet", the theme from Serial Experiments Lain, and "Obsession" from .hack//SIGN are good examples. (One reason production companies may be doing this is to shortcut around the tendency, noted above, for American importers to ditch Japanese theme songs.)
The songs also frequently follow a specific pattern: the first verse is usually about a minute along, and changes to a different tune around halfway through. There is an instrumental of a few seconds before the song repeats, with different lyrics (althrough some will stay the same). The song then has a longer instrumental (throwing lyrics in there isn't unheard of, but they are always different to the lyrics from both verses), before repeating the second part of the first verse. Some anime like to play with this format: for instance, Neon Genesis Evangelion's theme, "The Cruel Angel's Thesis", had the longer instrumental (with lyrics for a few seconds) between the first and second verses, and had the rest of the song (second verse, lyrics inbetween, and repeat of half of first verse) in quick succession.
This doesn't just apply to just anime. Various video games from Japan will also have anime theme songs that's used in openings, endings, cutscenes (best example being "Suteki da Ne" for Final Fantasy X), and commercials to promote both singer(s) and game. Usually, the song is released by the artists before the games are and have no actual ties to it, but occasionally the song will be written with the specific game in mind (a good example being Tales of Phantasia and Yume o Owaranai).
This page is example-less, as the trope is omnipresent. Potential sub-tropes are, naturally, a different story.
See also Image Song.