What does it mean to multiply an element of a group by a subgroup?

From my Discrete class notes:

Let $\displaystyle G$ be a group (a set with an operation that is closed and associative, has an identity and every element has an inverse). Let $\displaystyle H$ be a subgroup of $\displaystyle G$.

$\displaystyle \forall a,b \in G$, define relation $\displaystyle R$ as $\displaystyle aRb$ if $\displaystyle aH=bH$.

Relation $\displaystyle R$ is called a 'coset'.

What does that relation, coset, mean? I interpret the definition as "$\displaystyle a$ relation $\displaystyle b$ if $\displaystyle a$ left times a subgroup equals $\displaystyle b$ left times the same subgroup". But that is kind of meaningless. What is an element times a subgroup?

Thanks, in advance,

Jeff