FICO = Fair Isaac Corporation.
When you hear people talking about your credit score, FICO is the company that provides it. They calculate your credit score based off of the information in your credit report. There are three different bureaus that carry information about you: TransUnion, Experian, and Equifax (so you actually have 3 different credit reports - one from each bureau).
To be a true, actual credit score, it has to be a FICO score. Some other sites like Credit Karma and Credit Sesame, as mentioned above, have their own algorithms they use to estimate your credit score based on your credit report, but they are not official FICO scores. As the previous poster mentioned, if you have a credit card issued by Discover or Barclaycard US, you can get a TransUnion FICO from them. The Credit Sesame and Credit Karma scores are close enough to give you a good idea, though I feel like the score Credit Sesame gives me is quite a few points higher than my actual Experian score. YMMV.
Credit scoring is a strange beast. You actually have a lot more than one score, and even more than three scores. FICO has different scoring models based on what you kind of credit you're requesting, and even different models within those sub-categories. You actually have no way of knowing most of these scores other than if you apply for a loan and your lender tells you what score they got. Check out the following link for an interesting overview of what I summarized here:
http://www.bankrate.com/finance/credit/how-many-fico-credit-scores-do-you-have.aspx