The group SL(2,Z) is the special linear group of 2x2 matrices with integer entries and determinant 1. This group has a special property that makes it the outer-automorphism group of Z^2.
The outer-automorphism group of a group is a measure of its symmetry and structure. In the case of Z^2, the fact that its outer-automorphism group is SL(2,Z) tells us that Z^2 has a rich and interesting structure.
The modular group is a subgroup of SL(2,Z) and is generated by two elements, known as the matrices S and T. These two elements form a basis for the outer-automorphism group of Z^2.
One application of this fact is in the study of modular forms, which are functions that transform in a certain way under the action of the modular group. These forms have important applications in number theory, algebraic geometry, and physics.
Yes, there are other groups that have SL(2,Z) as their outer-automorphism group, such as the Heisenberg group and some other higher dimensional groups. However, the modular group is the most well-known and extensively studied example.