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Defensive tackles have two important jobs. First, their job is to make sure that the running back can't run straight up the field. If the tackles clog up the middle of the line, then the running back will have to go the long way around, which gives the rest of the defense time to catch him and tackle him. Second, if it's a passing play, their job is to push the center and guards backwards. This is because the quarterback will be stepping backwards, and there are defensive ends who will be running around and trying to get him from the side or from behind. To avoid these guys, the quarterback is going to want to step forward into the "pocket," the safe area held open by his center and guards. If the defensive tackles can close up this space, the quarterback will be stuck where he is and the defensive ends will have a much better chance of getting him.
There are a lot of different ways to line up your tackles, and they have different names depending on how you line them up. On the offensive line, there are gaps between the linemen. The first important thing for a tackle is how many gaps he has to cover. Sometimes the defensive scheme says he is responsible for only one gap - it's his job to make sure the running back can't come through his gap, and the other gaps will be someone else's responsibility. In this case we say the tackle is playing in a one gap defense. The tackle will line up right in the gap, not directly facing any offensive lineman.
In other schemes, the tackle will be responsible for two gaps. In this case the tackle will line up directly facing an offensive lineman, and his job will be to push that lineman backwards and make sure the running back doesn't run past on either side of his lineman. If you want to play a two gap scheme, you need larger stronger defensive tackles who can control an offensive lineman. If you want to play a one gap scheme you can use slightly smaller defensive tackles who are faster and more athletic and can penetrate into the offensive backfield more often. In a two gap scheme, the tackles are supposed to control the linemen, thus making sure that no one is blocking the linebackers behind them and the linebackers are then free to make the play and tackle the runner. So in a two gap scheme, you don't expect the defensive tackles to have a large number of sacks or tackles. They are doing their job if the linebackers have a lot of sacks and tackles.
In a one gap scheme, the defensive tackle is supposed to tackle the running back if he comes in the tackle's gap. On a passing play, the tackle is supposed to get into the quarterback's area and screw up the play, possibly tackling the quarterback for a sack. So you expect tackles in a one gap scheme to tackle the runner and sack the quarterback more often.
If you want to be especially good at rushing the passer you'll find four relatively athletic, perhaps somewhat smaller defensive linemen and line them up in a 1 gap scheme. Now you will have at least three gaps unprotected, so it's important in this scheme that you have three very solid linebackers who can cover these gaps.
If you want to be particularly good at stopping the run, then you will get four heavier and perhaps slightly slower defensive linemen, and play them in a two gap scheme. Now the offensive linemen will be all tied up with your linemen, clogging up all the interior running lanes and leaving your three linebackers free to roam for the running back.
Just as the offensive linemen have choreographed routines to block the defense, the defensive linemen have choreographed routines to try to get into the offensive backfield. Two of the most popular are called a stunt and a zone blitz. In a stunt, one lineman will block an offensive lineman diagonally, say to his left. Then a second defensive lineman who started on the first lineman's left will take a half step backwards, run quickly around behind the first lineman, and then try to run into the backfield in the hole the first lineman created in his right.
Another favorite dance is called a "zone blitz." In this scheme, one or two linebackers will rush on the same side of the center, perhaps the weak side away from the tight end. So you have a center, a guard, and a tackle trying to stop a defensive tackle, a defensive end, and two linebackers. This is almost impossible. However, when you do this you leave a big hole in your defense where the two linebackers ran away to get into the backfield. What the defense can do about this is to have the defensive tackle and perhaps defensive end drop back from the line and try to defend those "zones" that the linebackers just vacated. Of course defensive linemen are not the greatest guys in the world at pass defense, but the idea is that they only need to defend these areas for about two seconds, then the blitz should be hitting the quarterback.
The linemen on the offensive line line up a few feet away from each other. This leaves gaps between the linemen. These gaps are both lettered and numbered, as shown below. The gap between the guard and tackle is called the B gap. If you are a defensive tackle lined up in the B gap, but shifted over a bit towards the guard, you're called a 3-technique. If you were lined up in the same gap but shifted over a couple feet to line up on the tackle's shoulder, you would be a 4-technique. If you line up directly facing the center, you're called a nose tackle.
Nose tackles are two-gap players, and are typically very big and very strong men, usually 350 pounds or more and 6'5" or taller. These guys have the responsibility of clogging up the entire center of the field, of keeping the center and at least one guard busy, and thereby protecting their middle linebacker. The nose tackle will also be responsible in passing plays to push the center back towards the quarterback so that the quarterback cannot step up in the pocket and evade the rush of the defensive ends. A good nose tackle can be hit simultaneously by 650 pounds of center and guard and will not budge as much as one inch.
A 3-technique tackle lines up between the offensive guard and tackle. A 3-technique tackle is supposed to run through his gap immediately. He is a 1-gap player. His job is not to block or get tied up in a block, but rather to be athletic and get himself into the offensive backfield and disrupt their plans. Because of this a 3-technique tackle is a lighter more athletic guy than a nose tackle, typically weighing more like 290 to 300 pounds.