Let $G$ be a group, then you can get an homomorphisms
$$ f \colon G \to S(G)$$
where $S(G)$ is the group of permutations on $G$, defined as the mapping that send every $g \in G$ in the mapping $f(g) \colon G \to G$ such that for every $x \in G$ the equation
Associativity and unit axioms grant that $f$ is an homomorphism of groups:
- for all $g,h \in G$ and $x \in G$ we have
- for all $x \in G$ the equality
This homomorphism $f$ is the left regular representation (it represents the elements of the group $G$ as symmetries of the same group seen just as a set).
The name left is to distinguish from the (anti)-homomorphism
$$f' \colon G \to S(G)$$
that to every $g \in G$ associate $f'(g) \colon G \to G$ such that for every $x \in G$ we have $f'(g)(x) = xg$.
Via the correspondence between actions and homomorphisms in symmetric groups left regular representation are those representation that correspond to left-action of $G$ on itself given by multiplication.
Hope this helps.