an irreducible component

Some shapes come in obviously separate pieces: the equation xy = 0 in the plane is the union of the x-axis and the y-axis, two genuinely different lines crossing at the origin. The irreducible components of an algebraic set are its largest indivisible pieces — the natural way to decompose it into parts that cannot themselves be broken into smaller algebraic sets. For xy = 0 the two axes are the two components.

An algebraic set X is irreducible if it cannot be written as a union X = X_1 union X_2 of two proper closed subsets; otherwise it is reducible. The decomposition theorem says every algebraic set X is uniquely a finite union of irreducible closed subsets, X = Y_1 union ... union Y_r, where no Y_i is contained in another. These maximal irreducible closed subsets are the irreducible components, and the no-containment condition makes the list unique. Algebraically, this mirrors the primary decomposition of ideals: the components of V(I) correspond to the minimal prime ideals containing I, since irreducible closed sets correspond exactly to prime ideals under the Nullstellensatz dictionary. Finiteness of the list is guaranteed because k[x_1, ..., x_n] is Noetherian.

Decomposing into components is the first thing you do with any new variety, because most theorems are stated for irreducible varieties and you reduce to that case. Where it bites: a variety can have components of different dimensions glued together — for instance V(xz, yz) in A^3 is the plane z = 0 (a surface) union the z-axis (a line) — so dimension is the maximum over components, and 'the dimension of X' can hide that X is not pure-dimensional. Also, irreducibility is strictly stronger than connectedness: two crossing lines are connected but reducible, while a single irreducible variety is automatically connected.