In matrix notation let A be some matrix. The complex conjugate (c.c.) of A is denoted with a "*" i.e.

c.c.(A) defined A*.

You generate it by replacing each element of the matrix aij with its complex conjugate a*ij

In quantum mechanics wavefunctions can be represented in matrix form. The proabaility of a state is the integral over space of the wavefunction multiplied by its c.c. This gauranties a positive definite value for the probability.

A real number is equal to it's complex conjugate.