The gaps on either side of the center are the A gaps. The gaps between the guard and tackle are the B gaps. Outside the tackles are C through E gaps, but we'll concentrate on the first two.
A defensive lineman can do one of two things, he can line up (generally) over an offensive lineman, or he can line up in front of one of the gaps between offensive linemen.
If he is lining up directly over an O lineman, then he is responsible for controlling the gaps on either side of that lineman. His job is to drive into that lineman, maintain his position and then if the ball carrier comes into either one of those gaps on each side, slide into the gap and stuff the play. These guys can be called nose guards, nose tackles, 2 gappers. As you can imagine, a two gap DT must be bigger stronger, and extremely durable. You think of guys like Butz, Adams, Siragusa.
If he is responsible for a single gap, then his job is to prenetrate through that gap, get into the backfield and cause as much disruption there as possible. This requires more quickness and explosion at the snap. Think Sapp.
But it gets a little more complicated. Teams assign numbers to the OL, in fact, they can assign two numbers to each O lineman depending upon which shoulder (left or right) they want the D lineman to attack. They call these numbers 'techniques', although the term is a somewhat of a misnomer.
The 0 technique or position is right over the center's nose.
The 1 technique is lining up over the center's left or right shoulder (in the A gap).
The 2 technique is on the shoulder of the OG closest to the center, right shoulder for the left guard, left shoulder for the right guard, (in the A gap).
The 3 technique is the other shoulder (outside) of the OG, closest to the OT on each side (in the B gap).
The 4 technique is the OT's inside shoulder, closest to the center (in the B gap).
The 5 technique is the OT's outside shoulder (in the C gap).
Thus a 3 technique DT is a gap shooting tackle who specializes in the 'B' gap between the OG and the OT, with an alignment emphasis at the snap for being closer to the outside shoulder of the OG rather than closer to the OT. By lining up this way, he can shoot straight into the gap, or turn and trap the OG inside thus creating more of a lane for a blitzer.
At the same time, a nose tackle is typically asked to play the 0 technique, right over the center's nose, and control the gaps on each side of the center. At least one of the OGs will be doubling up on him each play, hoping that they can drive him the opposite way and crate a lane behind the driving guard, or lock on to him so that he can't slide into the other A gap.
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