So I promised I'd put up a blog post about the Gettier Problem in my Ancient philosophy class. So here it is.
The gist of the problem, is that Justified True Belief does seem necessary for knowledge (you need all three), but it isn't sufficient for knowledge (you can still not know something even if you have all three).
I think the easiest example of the Gettier problem to wrap people's mind around it, is the one I used in class. You walk into a room, and you see someone who looks like your friend Bob. You form the belief, "Bob is in the room." Its true. You believe it, and you're justified in believing it because you see someone who resembles Bob in the room. But who you see is not Bob, but Steve, Bob's twin brother. Bob is hiding under the bed.
Now the common objection is that my justification is based on something untrue.... But everything I've said is true... I've seen someone who looked like Bob.
In class I replaced Steve with a mirrored reflection of Bob. I see someone who looks like Bob, (but I'm not actually seeing Bob) etc. etc. In that circumstance lots of people want to say I have knowledge. But if thats the case, how is that meaningfully any different than the Steve case? In both cases I'm looking at something that resembles Bob, but is not actually Bob.
The wikipedia entry on the Gettier problem is pretty good, and covers most of the other common objections, and some of the less common attempts to solve the problem. Here you can read is actual paper. Again, I can only wish I could write such a short paper, and change philosophy so significantly.