Lie bracket
If two operations commute, doing them in either order gives the same result and there is nothing to measure. The Lie bracket is precisely the gadget that measures how badly two things fail to commute. Think of slightly rotating a book about its horizontal axis and then its vertical axis, versus doing it in the opposite order: the two final orientations differ, and that difference — to first order — is captured by a single new element, the bracket of the two rotations.
Formally the bracket is the bilinear, antisymmetric map [ , ] : L x L -> L that, together with the Jacobi identity, defines the Lie algebra L. Antisymmetry says [x, y] = -[y, x], so the bracket vanishes on equal arguments. When L sits inside an associative algebra A, the canonical bracket is the commutator [x, y] = xy - yx; for matrices this is XY - YX, which is zero exactly when X and Y commute.
A Lie algebra is called abelian when its bracket is identically zero — then it is just a vector space with no extra structure, the linear shadow of a commutative group. So the bracket is the entire content of the theory: knowing the vector space tells you nothing, knowing the bracket tells you everything.
In sl(2) with basis e = [0, 1; 0, 0], f = [0, 0; 1, 0], h = [1, 0; 0, -1], the brackets are [h, e] = 2e, [h, f] = -2f, [e, f] = h. Check [e, f] = ef - fe = [1, 0; 0, 0] - [0, 0; 0, 1] = h.
The three defining brackets of sl(2), the workhorse of the whole theory.
In differential geometry the same name is used for the bracket of vector fields, [X, Y]f = X(Yf) - Y(Xf), which makes the vector fields on a manifold into an (infinite-dimensional) Lie algebra. The matrix commutator is the finite-dimensional special case.