Classically, “Galois extension” refers to a class of extensions of fields.
An extension K⊂LK\subset L of fields is Galois if it is normal and separable. In that case, the automorphism group Aut K(L)Aut_K(L) of KK-automorphisms of LL is called the Galois group and often denoted by Gal(L:K)Gal(L:K).
There is a famous Galois theory for such extensions.
Now there are numerous generalizations, including Hopf-Galois extensions, coalgebra Galois extensions etc.