Cornerback

These guys are very fast. 
They are typically anywhere from about 5'9" to 6'2" and weigh about 180 to 200 pounds. They can run at near Olympic speeds. Their job is to stay right next to the wide receivers and prevent passes. Their favorite way to do this is to intercept any pass that comes their way. A typical cornerback will intercept about five passes a year. Some cornerbacks are so good at intercepting the ball that quarterbacks simply won't throw to any receiver that they are covering. Their next favorite is to bat the pass down just as it gets to the receiver. Failing that, they will try to push the receiver out of bounds just as or after he catches the ball, or tackle him. Most cornerbacks don't like to tackle very well. There's too great a chance to get hurt.
Cornerbacks and Safeties are called the Secondary. This is because they are the second line of defense, behind the defensive linemen and the linebackers. Running backs just love to get into the secondary. This means they're past the linebackers, so they have a big gain already. And cornerbacks don't hit anywhere near as hard as linebackers do.
There are a few different ways the cornerback can try to play his position. First, there's what is called "bump and run" coverage. In this type of coverage, the cornerback will block the wide receiver at the line, just as the play starts. If the cornerback can delay the wide receiver for as little as one second, there's a good chance that the timing of that wide receiver's route will be all screwed up and the quarterback won't be able to find him on the field. In the diagram above, the cornerback covering the split end likely will use bump and run coverage. However, the flanker is lined up a few yards behind the line, so he has a good chance of evading the cornerback on the right side. It's unlikely that the cornerback on the right will be able to use bump and run coverage. After the receiver has gotten five yards past the line, it's illegal to grab him, block him, bump him, or interfere with him. If the cornerback does, a referee will most likely notice it and issue a penalty. The penalty will be that the ball will be placed as if the receiver had caught it, whether the ball was thrown to him or not.
The cornerback may be in what is called man coverage. In man coverage the cornerback and the receiver are going to be all by themselves, one on one. If the cornerback is in man coverage, he will most likely try to funnel his receiver over to the edge of the field, to limit where the receiver can run. In the diagram above, the cornerback on the right is lined up a bit inside of the receiver, showing that he will be trying to get the receiver moving towards the outside of the field. This is a sign that this cornerback is in man coverage.
The cornerback may have "help over the top." This means that the safety on his side of the field will be staying close to him, and if the ball is thrown his way the safety will quickly come over to help out. There are some outstanding receivers in the NFL, and it is typical for these receivers to be covered by two guys in this fashion.
The cornerback may be in "zone coverage." This means the cornerback is responsible for an area on the field. If a receiver comes into his area, the cornerback will cover him. When the receiver leaves the cornerback's area, he will pass him off to a different cornerback. The single biggest advantage of zone coverage is that the cornerbacks play back a little bit and can keep facing the quarterback. This means they watch the quarterback continuously, and know instantly when and where he throws the ball. This gives them the best chance to intercept the ball. In man coverage the cornerback is running step for step with the receiver, so he cannot watch the quarterback closely to see exactly when and where he throws the ball.
Fans are often disappointed and upset when a cornerback on their team is in position to make an interception and drops the ball. This is actually not at all uncommon. First, the cornerback has no practice time with that quarterback and has no experience catching his passes. Second, as Chris Collinsworth has pointed out, "If they could catch the ball, they'd be receivers."
As the NFL adjusts its rules to become more of a passing league, the value of cornerbacks is going up. The best cornerbacks can make $10M per year.