The area on the field that extends laterally between the offensive tackles on either side of the offensive line. The tackle box is significant for both the offense and the defense.
For the offense, the tackle box is the area that defines certain rules for the quarterback. The quarterback cannot throw the ball away in an area not occupied by an eligible receiver when he is inside the tackle box.
For the defense, the numbers of defenders that are lined up inside the tackle box tends to denote how much the defense is committing to stopping the run. The tackle box tends to be quite similar to the hash marks, but can move a bit from side to side depending on where the ball is snapped from.
Also, on March 20th, 2013 the NFL passed a player safety rule that aims to prevent the ball carrier and defensive tacklers from leading with the crown of their helmet when both players are outside the tackle box.
The tackle box is a nice indicator for a football watcher to determine what is happening on the field. On offense, the tackle box represents the "pocket" that the quarterback stays in while protected by the offensive line. When the quarterback leaves the tackle box, he is no longer bound by most of the intentional grounding rules.
On defense, a team will load the box with defenders if they are particularly worried about a team's running offense.
A quarterback will generally look at the number of defenders in the box before determining whether or not to audible to a different play.