A real number is an algebraic number if it is a zero of a polynomial with integer coefficients; and its degree is the least of the degrees of the polynomials with it as a zero. For example, the rational number a/b (with a, b and non-zero integers) is an algebraic number of degree one, because it is a zero of bx-a. The square root of two is an algebraic number of degree two because it is a zero of x2-2.
If a real number is not algebraic, then it is a transcendental number. The base of the natural logarithms e (2.71828...), and π (3.14159....) are both transcendental. In fact, almost all real numbers are transcendental because the set of algebraic numbers is countable.