What is a coset? Honestly. I just don't understand the word and keep running into highly technical explanations (wikipedia, I'm looking at you!). If somebody would be so awesome as to explain the concept assuming basic knowledge of group theory and high school algebra I would be delighted.

If $H$ is a subgroup of $G$, then you can define a relation on $G$ by setting
$$
a\sim_H b\qquad\text{if and only if}\qquad ab^{-1}\in H
$$
It's a useful exercise in applying group laws proving that $\sim_H$ is an equivalence relation and that the equivalence class of $1\in G$ is
$$
[1]_{\sim_H}=H
$$
This equivalence relation has very pleasant properties; for instance, if $a\in G$, then there is a bijection $\varphi_a\colon H\to [a]_{\sim_H}$ given by
$$
\varphi_a(x)=xa
$$
(prove it). In particular, all equivalence classes are equipotent so, in the finite case, they all have the same number of elements. This bijection shows also that
$$
[a]_{\sim_H}=\{\,ha:h\in H\,\}=Ha
$$
the right coset (somebody calls this a left coset, check with your textbook) determined by $H$ and $a$.
This readily proves Lagrange's theorem: if we denote by $[G:H]$ the number of equivalence classes and $G$ is finite, we have
$$
|G|=|H|\,[G:H]
$$
because we can just count the number of elements in one equivalence class ($H$ or anyone else) and multiply by the number of classes, since they share the number of elements.
There is of course no preference for the right side; one can define
$$
a\mathrel{_H{\sim}} b\qquad\text{if and only if}\qquad a^{-1}b\in H
$$
and prove the same results as before, with the only difference that the map will be
$$
\psi_a\colon H\to[a]_{_H{\sim}},\quad \psi_a(x)=ax
$$
and the equality
$$
[a]_{_H{\sim}}=aH.
$$
In general the two equivalence relations are distinct. They are the same precisely when the subgroup $H$ is normal or, equivalently, one of them is a congruence, that is, for $\sim_H$, from $a\sim_H b$ and $c\sim_H d$ we can deduce $ac\sim_H bd$.
