Definition of a left coset? There is something that I do not understand about the definition of a left coset.

Let $\,G\,$ be a group and $\,H\,$ be a subgroup of $\,G\,$. Then the subset $\,aH=\{ah |h \in H\}\subseteq G\,$ is the left coset of $\,H\,$ containing $\,a.$

What is "$\,a\,$" in this definition? What does it represent? Can anyone help?
Thanks.

$a$ can be any element of $G$. For example, let's take the symmetries of the square, $D_4$, and a subgroup of it: $H = \{e, r_1, r_2, r_3\}$. We'll pick $f_v$ for our $g$. $gH = \{ gh : h \in H\} = \{ f_ve, f_vr_1, f_vr_2, f_vr_3 \} = \{ f_v, f_d, f_h, f_c \}$. $f_v$ turns the group of rotations into a set of flips (which is not a group).
