PSP doesn't have very many good games that I've seen my friend play (she has a PSP, I have a DSi). Her collection includes (or included, since there are some games she no longer has)
DIssidia Final Fantasy (Love it, very good game. Probably the best on the PSP.)
Fading Shadows (Puzzle. She hates it, but I don't see the big deal.)
Darkstalker Chronicles the Chaos Tower (Or something. good fighting game)
Final Fantasy 7 Crisis Core (X-Play gave it a 2/5. I told her and she pounded the hell out of me. She beat it, loved it, then it broke or something bad. She's over it now.)
Astonisha Story (Your basic run-of-the-mill RPG, but it takes a good 15+ hours to beat...I think. I wouldn't know, cuz it broke before she could beat it.)
Pangya (An anime-golf game where the characters take their golfing VERY seriously. So much so that in the opening movie, they all pour their spirits in a golf ball. Funny to watch. She says it's fun, but I actually read reviews and she doesn't...)
Now for the DSi, The games in MY arsenal include:
The World Ends With You (Hands down, THIS IS THE BEST DS GAME EVVVUURR!!! XD I'm a huge fan. It got a 5/5 on X-Play. I can tell you a whole essay on what I like about it, but I'll save that for when you ask me my personal opinion on it.) ...0_o *obsession* I went so far as to buy it TWICE (cuz I lost my first copy.) The ONLY game I'd ever done that for to date. Hoping for a sequal, but that's unlikely.
Castlevania: Portrait of Ruin (It was fun. I wanna play Dawn of Sorrow cuz people said it was the best DS castlevania game ever. DoS 5/5 by X-Play. PoR got a 4/5.)
Sonic Rush Adventure (Actually a good Sonic game in EVUR!!! Good like the Advanced series, and side quests were plentiful. Better than UNLEASHED in most ways. 4/5 by X-Play.)
Bowser's Inside Story (Meh...it was a fun game for 25+ hours, but I wouldn't replay the game again if I had a choice, since I can do all the sidequests(?) while playing the main game. Plays kinda like Paper Mario, and it has its funny moments. X-Play gave it a 5/5. I think it's average.)
Lunar Knights (Beat the main game in 15+ hours. At the time, I was obsessed with everything Boktai, and logged 99:59+ hours on it. unlocked everything. Personally think that Boktai 2 Solar Boy Django for the GBA was a better game, cuz it was longer and had more bosses.
Kingdom Hearts: 358/2 Days (A KH game...WAY more better than Chain of Memories, but unfortunately took less time to beat (still stuck on Marluxia in CoM. ugh.) unlocked everything 100%, and haven't picked it up since. now it's $20 at gamestop...I feel cheated. RE: Coded had BETTER be worth my time AND money!!! XP)
Also, I'm looking forward to Kingdom Hearts RE: Coded, Sonic Colors, Golden Sun: Dark Dawn, the new Nintendo 3DS, and Okamiden coming out in Oct. 2010 - May. 2011
Obviously, the DS has a larger game library and more memorable titles (Mario, Sonic, Zelda, etc.) than the PSP. But if you do get it, the Internet is slow and you can't watch videos. To listen to music, you have to convert the mp3 files to either m4A or AAC files, which is a hastle unless you have iTunes. (sad. hope the Nintendo 3DS fixes ALL these problems and more!!! DX) Get this if you're a party/RPG gamer. I'm sure it can take a beating better than a PSP if it's dropped on its side.
The PSP has alot of sucky 3rd party games, and the few good ones (God of War, Dissidia FF, games that made you wanna play them CONSTANTLY, not b/c they were the only ones you had) take a back seat to the other piles of dreck. (Ben 10, Lego whatevers, etc.) However, you can listen to mp3s, look at youtube videos, and surf the Internet faster and more efficently than the DSi. (GRR!!! XP) Be careful, cuz the UMD disks are easier to break, so don't hand 'em to kids who ask to play. Get it if you're into single-player games and want...better graphics in games (I can't honestly tell you about that, cuz if the plot, gameplay and hours are absolutely perfect, the graphics can go out the door!) Also, I'm more afraid to drop a PSP on the floor than a DSi, cuz I'm sure Japanese-manufactured parts will fly EVERYWHERE!!!!! X(
Just my honest opinions and experiences about both game systems. Have fun with whatever you pick.