Euclidean domain
A Euclidean domain is an integral domain in which you can still do long division with a remainder that is genuinely 'smaller' than the divisor — exactly the engine that makes the Euclidean algorithm for greatest common divisors run. Wherever you can divide-with-remainder, you can compute gcds, and a great deal of clean structure follows for free.
Formally, an integral domain R is Euclidean if there is a function (a Euclidean norm) N from the nonzero elements of R to the nonnegative integers such that for any a and any nonzero b in R, there exist q and r with a = b·q + r where either r = 0 or N(r) < N(b). The norm measures size and the inequality guarantees the remainder shrinks, so repeated division terminates. There is no uniqueness requirement on q and r.
Euclidean domains sit inside a strict hierarchy: every Euclidean domain is a principal ideal domain (the smallest-norm nonzero element of an ideal generates it), and every PID is a unique factorization domain — but neither arrow reverses. The ring Z[(1+√−19)/2] is a famous example of a PID that is not Euclidean, showing the Euclidean condition is strictly stronger than being a PID.
Z is Euclidean with N(a) = |a|. The polynomial ring k[x] over a field is Euclidean with N(f) = deg(f). The Gaussian integers Z[i] are Euclidean with N(a + bi) = a^2 + b^2.
Three Euclidean domains with their standard norms.