A function made up of a finite combination of constant functions, field operations (i.e. addition, multiplication, division, and exponentiation), and algebraic functions. It also includes the transcendental function: exp(x) and all of their inverses. This includes logarithmic functions (inverses of exponentials) the trigonometric functions (as they can be expressed as linear combinations of exponential functions with imaginary arguments), and the hyperbolic functions (as they are also linear combinations of exponential functions]).
Not all functions are elementary, of course. The integrals of some elementary functions are impossible integrals because they cannot be expressed in terms of elementary functions. Infinite series of elementary functions are also in general not elementary.