Here's some of my favorites of the past few seasons. Some of these will be repeats of what people here have already said.
Birdy the Mighty was a four-episode OVA (direct to video) release in the 1990s but got revived as a two-season TV series (so far) in 2008 and 2009. Birdy Cephon Altera is, for all intents and purposes, a "space cop" going after various criminals. In her pursuit of them to Earth, she accidentally kills a high school boy. In order to save his life while her people's technology rebuilds his body, she agrees (or more correctly is ordered by her bosses) to host his consciousness within her body. The story continues with the two of them sharing one body, able to take either his or her form depending on what the situation needs, and trying to balance his school and personal life with her Federation duties (and eventually her personal life as well). Besides the humor of the whole situation, the storyline of the pursuit of the criminals is well-thought-out and captivating - especially in season 2.
Black Lagoon is, as mentioned above by someone else, a fun, bad-ass anime. A meek, 20-ish Japanese salaryman, feeling like he's at a dead-end in his career path, is sent by his boss to southeast Asia with a disc full of company secrets. A group of American mercenary pirates based in Thailand, consisting of a Vietnam vet deserter, a Kevin Mitnick-type hacker on the run from the FBI and the mob, and a gunslinger fleeing a life of abuse in NYC, abduct him and steal the disc. His company decides the disc is so important that none of them can live, and send a mercenary army out to take them all down. They of course survive, and the salaryman, fed up with his life, throws in with the pirates, starting the series' storyline. There have been 2 seasons so far and a third OVA release due to come out in 2010.
Full Metal Panic I will mention simply because if you have only seen the first season (and not Fumoffu? or The Second Raid), you're really missing out. Fumoffu? is there for the funny side of things taking place mostly at Jindai High, while TSR picks up the main storyline back with the mercenary army. Fans are hoping that the series will eventually continue - there is a ton of story left in the original novels that takes you on a wild ride and makes some abrupt, hard, turns here and there to shake up the cast you know and love.
Girl Who Leapt Through Time is a movie-length OVA which is pretty much summarized in the title - it's about a girl who learns how to travel very short distances (hours, maximum) back and forth through time. In Japan, OVA releases tend to be higher quality, especially one-offs like this one, and this show very tidily encapsulates the entire story it's telling. It's one of the best fully-contained-in-itself stories you can watch.
K-ON! is a good fun "slice of life" show for those who might get their fill of action/adventure/mecha anime. It's about four (eventually five) girls who form a band in high school. There's not much else to say.. the "slice of life" genre has been compared to Seinfeld in the past, as in, it's a lot of stories about nothing in general.
Kyouran Kazoku Nikki is a show about a paranormal investigator who is given the task of finding a creature who every 1000 years is destined to destroy all life on Earth. Investigations has pinned it down to as close as one of 4 or 5 creatures that may or may not have the demon dormant within them. The solution is "Operation Cozy Family" where they all band together and act as a family unit, with a mother coming from a parallel dimension where she's a demi-god princess, a son which is a king of lions (literally), a jellyfish-like younger daughter, an ordinary human older daughter, and some others thrown in for good measure.
Macross Frontier is decent if you have been following various other Macross shows. Similar formula as other series - fighter pilot fighting against aliens determined to extinguish the human race exploring the galaxy, love triangle, etc. Good music and good blend of CGI and regular animation in this one.
Gotta stop here to head to work. Will add more later - only about halfway through my favorites list.