The center of the chess board refers to the four middle squares of the chessboard. In algebraic notation, they are the d4, e4, d5, and e5 squares.
Controlling those squares, directly or indirectly, usually means controlling the most active part of the chessboard. From the center, one has the most space to control the rest of the chessboard.
In the opening, most chess pieces will move through the center for attacking or defending. All the pieces have more squares to attack (or defend) when in the center.
A Knight in the center has eight squares it can cover. A Knight on the edge of the board only has four squares to cover and if the Knight is in a corner, it only has 2 squares in can cover.
A Bishop can cover 13 squares in the center but only 7 squares in the corner.
You should always try to develop your pieces to control the center. That can be done directly with the pawns on d4 and e4 (or d5 and e5 for Black), or indirectly in hypemodern fashion with the Bishops fianchettoed to the b2 and/or g2 squares and controlling the long diagonal (a1-h8 or h1-a8).
Center control is when the pieces are aiming at the four center controls. And greater center control leads to greater piece mobility with better chances for attacking or defending. The loss of the center usually means a cramped and restricted game. Controlling the center or losing control of the center in the opening could lead to a short game.
Here are some example traps based on control of the center or loss of control of the center.