Every subgroup of a group has left cosets (True or False) This is true, because one can always multiply elements of $G$ with elements of $G$ (in particular, the ones that are contained in $H$), and that is a non-empty subset of $G$, because G is a group.
Is this correct?
 A: As I see it, the question is "under which condition on the subgroup $H\leq G$ is the following relation : $g_1Rg_2$ iff $g_2^{-1}g_1\in H$ an equivalence ?"
If it is an equivalence relation then one can easily see (using the fact that $H$ is a group) that equivalence classes of $R$ are of the form $gH$ where $g\in G$. 
The main point here, is that this is always true (there are little things to write down but this is straightforward). In particular one does not need $H$ to be normal in $G$ to define such thing (students sometime make this mistake). 
The goal is probably to emphasize that we can always define left cosets for any subgroup $H$ of $G$ (leading for instance to Lagrange's theorem) and that normality comes into play only when we need to put a natural group structure on $G/H$. The word natural meaning that $g_1Hg_2H$ should be $g_1g_2H$ (it is not so hard to see where it comes from, natural means here that we want the natural projection of $G$ onto $G/H$ sending $g$ to its class to be a group morphism) and this last part is true for any $g_1,g_2\in G$ if and only if $H$ happens to be normal in $G$. 
