It's a number that isn't a root of any polynomial with rational coefficients, if I remember correctly. The square root of 2 isn't transcendental because it's a root of x^2 = 2. Not sure if complex numbers, quaternions, etc. can be transcendental...

A polynomial is the sum of a number of "terms" where each "term" is of the form
[coeficient * (base1 ^ exponent1) * (base2 ^ exponent2) etc. the coefficient and the exponents are constants, the bases are variables.)

It's easier to give examples of polynomials than to define them:

Some binomials (polynomials of two terms) include 2x + y, 5x² - 2x, 15y - 7, 4x²y - 3xy², and ax + b

Some trinomials (polynomials of three terms) include x² - 2x +1, 5x + 6y + 3, and ax + by + cz

A trancendental number is a number that either cannot be expressed as a polynomial(or its root), or only as a polynomial(or its root) which has an infinite number of terms.

Of the three most famous non-trivial irrational numbers, pi and e are transcendental, while phi (the golden mean) is not, since it equals 1/2 + √5/2

The sequence of types of numbers goes

Integers

Rational Numbers (integers plus fractions)

Real numbers (rational and irrational numbers, including transcendentals)

Complex numbers (reals, imaginaries, and vector sums of a real and an imaginary part)
[Imaginary numbers are the product of a real absolute value and ±i, the square root of -1]