This tag is for questions about algebraic groups. There are two main types of algebraic groups: linear algebraic groups and abelian varieties. The prototypical example of a linear algebraic group is $$\mathrm{GL}_n$$, the group of $$n\times n$$ matrices. The prototypical example of an abelian variety is an elliptic curve, which is the set of solutions to an equation $$y^2 = x^3 + Ax + B$$.
Over a field $$k$$, an algebraic group consists of (i) an underlying set $$G$$ defined as an algebraic subset of either affine space $$G \subset \mathbb{A}^n_k$$ (in the case of linear algebraic groups) or projective space $$G \subset \mathbb{P}^n_k$$ (in the case of abelian varieties) and (ii) a group operation called multiplication, which is a polynomial function $$m\colon G \times G \to G$$ satisfying axioms of associativity, invertibility, and identity. The set $$G$$ is referred to as an algebraic variety, and it is endowed with the Zariski topology, which is defined as the coarsest topology such that all subsets $$Z$$, which are cut out by the vanishing of a collection of polynomials, are closed. These closed subsets $$Z \subset G$$ are also algebraic varieties, called sub-varieties. If $$Z$$ is closed under the restriction of the multiplication map, i.e. if $$m(Z \times Z) \subset Z$$, then $$Z$$ also inherits a group structure and is called an algebraic subgroup of $$G$$.
Note that here we have not required a variety to be irreducible, which is a condition used by some authors. We have also not required $$k$$ to be algebraically closed, as some authors do.