Here are some terms that are non-trivial.

Congruence Subgroup
Any subgroup of the modular group which contains some principal congruence subgroup.
Elliptic (of matrices)
A Möbius transformation whose fixed points are exactly two distinct complex conjugate points.
Modular Group
The set of 2×2 matrices with integer entries and determinant 1. In many applications elements of the modular group are identified with their negatives and/or determinants of -1 are allowed as well.
Principal Congruence Subgroup of Level m
The subgroup of the modular group whose elements are all congruent to the identity modulus some natural number n. Note that the principal congruence subgroup of level 1 is the entire modular group.

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