A square number is the result of multiplying a whole number by itself. A cube number is the result of multiplying a whole number by itself three times.
Square numbers: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, \ldots$ $$n^2: \quad 1^2=1,\ 2^2=4,\ 3^2=9,\ \ldots,\ 12^2=144$$
Cube numbers: $1, 8, 27, 64, 125, \ldots$ $$n^3: \quad 1^3=1,\ 2^3=8,\ 3^3=27,\ 4^3=64,\ 5^3=125$$
Square roots and cube roots: $$\sqrt{49} = 7 \qquad \sqrt[3]{27} = 3$$
Every positive number has two square roots: $\sqrt{25} = \pm 5$, but by convention $\sqrt{25} = 5$ (the positive root).
Recognising them quickly is essential for factorising, Pythagoras, and indices.
Common error: confusing $3^2 = 9$ with $3 \times 2 = 6$ — squaring means multiplying by itself, not by 2.