If A is an Abelian group, a character is a group homomorphism into the multiplicative group of complex numbers. See also Dirichlet character.
If f is a representation of a group G, then the character of the representation is the function from G to the complex numbers given by the trace of f.
If A is an abelian algebra over the complex numbers, a character of A is an algebra homomorphism into the complex numbers. If in addition, A is a *-algebra, then a character is a *-homomorphism into the complex numbers.