Summary: If you love Chess and don't mind squinting at tiny screens, this is for you. The Chessmaster series provide a good way to learn to play, learn the fundamentals, practice to a high/intermediate level, and get a lot of useful exercises under your belt, if you have the determination to go through a whole lot of drilling. It isn't as feature-rich as the stationary console versions, but it will give back what you put into it if you aren't a grandmaster already.
Summary: I have personally gone through periods in my life where I love playing Chess and then other times I don't play it all but have always had a key interest in tactical games like this even though I've never been that great at them. It always amazes me when you watch the pros on TV playing for hours and hours. If you get Chessmaster on the PSP don't expect an eventful action game as that's not what it's about.