What is the difference between algebraic sets and algebraic varieties?

Question

I am wondering what is the difference between algebraic sets and algebraic varieties in complex projective space.

It seems that both are zero sets of polynomials, so what is the difference?

Answer 1

According to Wikipedia, some sources require that a variety be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. In this case, a non-irreducible algebraic variety is called an algebraic set. In other sources, this convention is not followed and both cases are simply called variety.

Answer 2

If we give $\mathbb A^n$ (resp. $ \mathbb P^n$) their Zariski topology, an algebraic set is just a closed subset $V\subset \mathbb A^n$ (resp. $V\subset \mathbb P^n$).
An algebraic variety is a vastly more general general concept, the basic object in classical algebraic geometry (even if since Grothendieck we have the even more general notion of scheme).
Every algebraic set, which a priori is a topological subspace, can be endowed with the structure of algebraic variety: the supplementary datum consists of decreeing which functions on open subsets $U\subset V$ are considered acceptable, thus obtaining the ring $\mathcal O_V(U)$ of "regular" functions on $U$.
By using this procedure we get the affine (resp. projective varieties).
However there are many algebraic varieties which are neither affine nor projective, and thus are completely different from algebraic subsets: the simplest example is the plane with the origin removed, $V=\mathbb A^2\setminus \{(0,0)\}$.