algebraic structure
Think of an algebraic structure as a set of objects together with a fixed list of ways to combine them, each combination obeying some written-down rules. A group, a ring, a vector space, a lattice — they all look very different on the surface, but each is just a set plus some operations plus some axioms. Universal algebra is the discipline that steps back and studies them all at once, asking which theorems are really about ‘groups’ and which are about ‘any set with operations satisfying equations’.
Formally, an algebraic structure (or simply an algebra) is a pair (A, F) where A is a set called the carrier (or underlying set) and F is a family of finitary operations on A, that is, functions f: A^n -> A for various non-negative integers n called arities. A nullary operation (n = 0) just picks out a distinguished element, like the identity of a group. The list of operation symbols and their arities is the signature; the equations or other formulas the operations must satisfy are the axioms.
Crucially, in pure universal algebra there are no relation symbols and no partial operations: everything is a total, everywhere-defined operation. Fields, for instance, sit awkwardly here because inversion is undefined at 0, so the class of fields is not a variety. This narrowing is the price of generality: by insisting on total operations and equational axioms one gets sweeping structural theorems (homomorphism theorems, Birkhoff's theorem) that hold uniformly across groups, rings, lattices, Boolean algebras and more.
A group is the algebraic structure (G, ·, ^{-1}, e) with one binary, one unary and one nullary operation, satisfying associativity, x·e = x = e·x, and x·x^{-1} = e = x^{-1}·x.
Including inverse as an explicit unary operation makes groups an equational class.
Beware the clash of conventions: in universal algebra ‘algebra’ means any set with operations, while elsewhere ‘algebra over a field’ means a vector space with a bilinear product. They are related but not the same notion.