In any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: $$a^2 + b^2 = c^2$$ where $c$ is the hypotenuse — the side opposite the right angle, always the longest side.
Finding the hypotenuse: $c = \sqrt{a^2 + b^2}$
Finding a shorter side: $a = \sqrt{c^2 - b^2}$
The 3-4-5 triangle is worth memorising ($9 + 16 = 25$). Other common Pythagorean triples: 5-12-13, 8-15-17.
Pythagoras also works in 3D — find a diagonal of a cuboid by applying the theorem twice, or use $d = \sqrt{a^2 + b^2 + c^2}$ directly.
Common errors: subtracting instead of adding when finding the hypotenuse, or forgetting to square-root at the end. Also: $c$ must be the side opposite the right angle — students sometimes pick the wrong hypotenuse.