algebraic number
Think of all the numbers you can ever pin down by writing a polynomial equation with whole-number (equivalently rational) coefficients and saying “this number is a solution.” The square root of 2 is one: it solves x^2 - 2 = 0. The golden ratio, the imaginary unit i, the cube root of 5 — all of them are caught in such a net. An algebraic number is exactly a number caught in some net of this kind; the numbers that escape every such net are called transcendental.
Precisely, a complex number a is algebraic if there is a nonzero polynomial f with rational coefficients such that f(a) = 0. Equivalently one may demand integer coefficients, since clearing denominators changes nothing. Every algebraic number has a unique monic polynomial of least degree over the rationals that it satisfies, its minimal polynomial; the degree of that polynomial is called the degree of the algebraic number.
The algebraic numbers form a field: sums, products, and inverses of algebraic numbers are again algebraic, and this field is the algebraic closure of the rationals inside the complex numbers. It is countable, so “most” complex numbers are transcendental — yet proving a specific number like pi or e transcendental is hard. Rational numbers are exactly the algebraic numbers of degree one.
The number sqrt(2) + sqrt(3) is algebraic of degree 4: it is a root of x^4 - 10x^2 + 1 = 0, and no rational polynomial of lower degree kills it.
Its minimal polynomial has degree 4 because the field it generates over the rationals has degree 4.
Being algebraic is a property of a number relative to a base field; here the base field is the rationals. Over a larger base field, more numbers become algebraic — every complex number is algebraic over the reals, since it satisfies a degree-two real polynomial.