A variety is an equationally defined class of algebras. As I understand it equationally defined means defined by universally quantified equations, for example the variety of all semigroups could be defined as the class of algebras fullfilling $x(yz) = (xy)z$.
But what about groups, there defining equations need existential quantification, the standard equations being: An algebra $(G, \cdot, 1)$ is a group iff
1) $\forall x,y,z : x(yz) = (xy)z$
2) $\forall x : 1\cdot x = x \cdot 1 = x$
3) $\forall x \exists y : x\cdot y = y\cdot x = 1$
The last equation (existence of inverses) I see as problematic, because it is not an universally quantified expression. So then group would not form a variety because they could not be defined by universally quantified equations, but in this article on the Variety of groups the opposite is claimed.
[...] Any variety of groups other than the variety of trivial groups and the variety of all groups [...]
So what did I missed here?
We introduce a fourth operation: $\neg$, and use the equation $$\forall x : x\cdot (\neg x) = 1 = (\neg x) \cdot x $$
Defining a new operation like so gives us the existence of an inverse without having the 'there exists' quantifier.