Galois extension
Think of a Galois extension as a field extension that is maximally symmetric: it has just enough room for every root that its polynomials demand, and no awkward roots that secretly coincide. When both of those good behaviours hold at once, the symmetry group of the extension is as large as it possibly could be, and that symmetry becomes a powerful tool for understanding the field.
Precisely, an algebraic extension L/K is a Galois extension if it is both normal and separable over K. Normal means that whenever an irreducible polynomial in K[x] has one root in L, it splits completely in L; separable means no irreducible polynomial over K has a repeated root in L. Equivalently, L/K is Galois exactly when the fixed field of the automorphism group Aut(L/K) is precisely K, and (for finite extensions) exactly when the number of K-automorphisms of L equals the degree [L : K]. The group Aut(L/K) is then called the Galois group, written Gal(L/K).
A caveat worth internalizing: in characteristic 0, and over any finite field, separability is automatic, so Galois just means normal there. The subtlety lives in positive characteristic with imperfect base fields, where inseparability can break the count [L:K] = |Gal(L/K)|. Also, 'Galois' is a property of the pair L/K, not of L alone: a field can be Galois over one base and not over another lying between.
Q(sqrt(2), sqrt(3)) over Q is Galois of degree 4, with Galois group Z/2Z x Z/2Z (each generator flips one of the two square roots).
A small biquadratic Galois extension with the Klein four-group as Galois group.
Some authors require finiteness to use the count |Gal(L/K)| = [L:K]; for infinite algebraic Galois extensions the right tool is the topology on Gal(L/K) as a profinite group and the Krull form of the Galois correspondence.