A priori knowledge is what you know without having to experience anything whereas a posteriori knowledge is knowledge derived from experience (empirical evidence).
How can anyone really know anything? That's a difficult question that philosophers have pondered since the ancient Greeks. Well, probably before then, but you don't get credit for anything unless you write it down for future generations to know. Even then, if you used paper you had better hope there are monks out there that want to duplicate your stuff until the printing press comes along.
Back in college I loved thinking about this sort of thing. I took a class in Philosophy of Science (I highly recommend that class for every person who intends to vote or think.) and enjoyed it so much that I got a minor in it.
Philosophy is basically math without numbers. It's where you go when you want to get meta about anything. If you want to know the atomic weight of calcium then you should ask a chemist or a physicist. If you want to know whether to believe the chemist or physicist then you go talk to a philosopher.
Quis custodiet ipsos custodes? (or would it be Quis custodiet philosophis dixit?) Well, if you want to know whether or not to believe a philosopher, you'll just have to ask other philosophers until you are satisfied. If you're still not satisfied, you can always ask a politician. They're always happy to tell you what you want to hear.
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