Linear algebraic group

In terms of underlying variety
A linear algebraic group or affine algebraic group is an algebraic group where the underlying algebraic variety is an affine variety.

By a basic theorem of algebraic geometry, any affine algebraic geometry has a faithful linear representation, and can hence be realized as a linear algebraic group. Thus, we often view affine algebraic group and linear algebraic group as synonyms.

In terms of embedding into general linear group
A linear algebraic group or affine algebraic group over a field $$k$$ is defined as an algebraic group $$G$$ such that there exists an embedding of $$G$$ as a closed subgroup of the general linear group $$GL(n,k)$$ for some choice of $$n$$ (and this embedding is a morphism of algebraic varieties).

Note that such an embedding as a closed subgroup of $$GL(n,k)$$ automatically gives an algebraic group structure, so if we provide an embedding, we do not need to specify an algebraic group structure separately. This is because $$GL(n,k)$$ comes naturally equipped with an defining ingredient::algebraic group structure, and any closed subgroup of algebraic group inherits algebraic group structure.