Lie algebra
Imagine you have a smooth group of symmetries — rotations of a sphere, say — and you want to study it not all at once but only near the identity, where the motions are tiny. Differentiating the group at the identity flattens the curved space of symmetries into a vector space, and the failure of two tiny symmetries to commute survives as a new operation called the bracket. A Lie algebra is exactly this linearized shadow: a vector space carrying just enough algebraic structure to remember how the original symmetries fail to commute.
Precisely, a Lie algebra over a field k is a k-vector space L equipped with a bilinear map [ , ] : L x L -> L, the bracket, satisfying two axioms: it is alternating, meaning [x, x] = 0 for all x (which forces antisymmetry [x, y] = -[y, x]), and it obeys the Jacobi identity [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0. Note there is no requirement of associativity or of a multiplicative identity; the bracket is generally neither.
The motivating example is any associative algebra A made into a Lie algebra by setting [x, y] = xy - yx, the commutator. In particular the n-by-n matrices under the commutator form the Lie algebra gl(n, k); its trace-zero part is sl(n, k). Over R or C, every finite-dimensional Lie algebra arises (via the Lie group correspondence) as the tangent space at the identity of a Lie group, and the bracket records the second-order non-commutativity of the group law.
Take L = R^3 with the cross product as bracket: [u, v] = u x v. It is bilinear, antisymmetric, and satisfies the Jacobi identity, so (R^3, x) is a 3-dimensional real Lie algebra — in fact isomorphic to so(3), the rotations of space.
The cross product is the everyday Lie bracket of the rotation algebra so(3).
Despite the name, a Lie algebra is rarely an associative algebra. The word algebra here just means a vector space with a bilinear product; the product happens to be the bracket, which is antisymmetric, not commutative.