An algebraic number is a root of a polynomial with integer coefficients (or, equivalently, with rational coeffients).
A number (especially a complex number) which is not algebraic is called transcendental; famous examples are the base ($\mathrm{e} = 2.7\ldots$) and period ($2 \pi \mathrm{i} = 6.28\ldots \mathrm{i}$, or equivalently $\pi = 3.14\ldots$) of the natural logarithm.
An algebraic integer is a root of a monic polynomial with integer coefficients. Given a field $k$ the (algebraic) number field $K = k[P]$ over $k$ is the minimal field containing all the roots of a given polynomial $P$ with coefficients in $k$. Usually one considers algebraic number fields over rational numbers.
algebraic number, algebraic integer