That's called the limited third-person viewpoint. Personally, I find it the easiest to pull off, and it seems to be the most commonly used one, but that's just me. (Hey, I write nearly all my stories with limited third-person. Lots and lots of writers I know use it, too. So...) It strikes the perfect balance, I think, between the intimacy of the first-person and the alacrity and flow of omniscient third-person.
First-person is nice for easier identification/relation/empathy with the character, but overdoing things is very easy. Melodrama, stream-of-consciousness and angst are the prime horrors lurking in this technique. Another is self-insertion: start using "I" to narrate things, and you run the risk of using the character as a mouthpiece rather than letting him/her have develop a personality. Symptoms include using language that you, the author, would use instead of one the character would use, or unconsciously turning the character into an outlet for your opinion. This is particularly dangerous when you feel the burning need to shoehorn a "meaningful" or "profound" theme into your story. (Which, by itself, deserves an entirely separate rant, but that's neither here nor there.)
I loathe the omniscient third-person perspective, and with good reasons. It gives the author an excuse to hop from head to head (which may just ruin suspense and intrigue, among other things), as well as dump information. Fantasy/sci-fi writers seem to fall into this trap a lot: shoving clinical, dull diatribes down the reader's throat. Really, if you absolutely have to include information about this race, that city, or this system of magic, then by all the good deities, do it in appendices! Under no circumstance should the reader be forced to go through expository paragraphs that read like a tourist brochure or an academic essay. It's one of the things that'll make me chuck a book across the room faster than you can say "JRR Tolkien."
