What is the significance of cosets in group theory? Cosets can be either left or right.  From what I understand left/right cosets partition a group.  Why is partitioning so important?
 A: The most important case is if you divide by a "normal" subgroup, in which case the left- and right-cosets coincide. Because then these cosets themselves form a group (called "factor" or "quotient" group).
More generally, it is often useful to consider how one group acts on the set of cosets (i.e. permuting them). You can often do some simple combinatorical arguments relating the size of groups, and the number of cosets and such things.
A: There exist numerous examples of the following situation: Suppose a group $G$ acts on a set $X$ of objects (e.g. on the left). This means that $\forall g \in G$ we have a bijection $\mu_g: X \rightarrow X$ in such a way that $\mu_{gh} = \mu_g \circ \mu_h$. Fix a point $x \in X$, then the set $O_x = \{\mu_g(x) | g \in G\}$ is called the orbit of $x$ under the action of $G$. Let the subgroup $H$ be the set of elements $h$ of $G$ that satisfy $\mu_h(x) = x$, this is called the stabilizer  subgroup of $x$ under the action of $G$. The point in this construction is that $O_x$ can be identified with the set of left cosets of $H$ in $G$. Let $x$ correspond to the coset of the identity (i.e. $H$), then we let any other point of $O_x$, say $y = \mu_g(x)$ correspond to the left coset $gH$. It is not hard to show that $\mu_g(x) = \mu_{g'}(x) \iff g$ and $g'$  belong to the same coset. So saying two points of $O_x$ are distinct correspond to saying that two cosets are disjoint and that saying that $O°x$ is the union of it's singletons is equivalent tot saying that $G$ is the union of the cosets. You can find more information here.
