Cosets Intuitively Explained I'm having a bit of trouble understanding what cosets actually mean, and the textbook (Fraleigh - Abstract Algebra) doesn't really seem to help much either. 
I know that the formal definition of a coset is:
$aH$  = $\{ah | h \in H\}$ = the left coset of H.
$Ha$ = $\{ha | h \in H\}$ = the right coset of H.
What do these "actually mean"? Any kind of "dumbed down" explanation would be tremendously helpful. Thank you.
 A: Let $H$ be a subgroup of a group $G$. Define an equivalence relation on $G$ by $x\sim y$ iff $x^{-1}y \in H$. The left cosets of $H$ are just the equivalence classes of $G$ under this relation (try to prove this). Right cosets are the equivalence classes of a similar relation ($x\sim y$ iff $xy^{-1}\in H$). An interesting and important fact about this equivalence relation is that all the equivalence classes (i.e., left cosets) have the same size, equal to the cardinality of $H$. Also note that $H$ itself is one of these equivalence classes.
Think about it this way. When you form a left coset, you're forcing $x$ to be the same as $y$ if $y=xh$ for some $h
\in H$. In other words, you're forcing elements of $H$ to act as right identities.
Eventually you'll learn that if $H$ is a "nice" type of subgroup called a normal subgroup, the cosets of $H$ are the elements a new group called a quotient group, in which $H$ is the identity element. This is what's really important about cosets. 
A: Not so much an explanation, as a couple of familiar examples that may help: 
The plane is a group. A line $L$ through the origin is a subgroup. A line parallel to $L$ is a coset of $L$. 
The integers are a group. The multiples of any fixed integer $n$ form a subgroup. The arithmetic progressions with common difference $n$ are the cosets of this subgroup. 
A: I think it's really helpful to go up a level and think of what equivalence classes are, because cosets are just a special case.
Whenever you have an equivalence relation on a set, it partitions the set (divides it up into nonoverlapping pieces) of things that are mutually equal. For each one of these pieces, we think of all the items collectively as one big point in a new set (the set of equivalence classes.)
For groups (an analogously for other algebraic objects), each coset is just a piece of a partition of the group defined using the operation in the group. Everything else mentioned above is still true (the cosets are "big points" in the set $G/H$. Each piece of the partition of $G$ has coalesced into a point of $G/H$.
Because of the definition of the equivalence relation used for cosets of a group, one of the cosets is always the subgroup itself. Everything else is a "parallel slice" of $G$ that has exactly the same number of elements as $H$. (That is a difference from general equivalence relations: the fact that the cosets all have the same size is a direct result of defining the group cosets. In general, equivalence class sizes could vary.) 
Finally, of course, when $N$ is a normal subgroup, those big points in $G/N$ have a natural group operation. That's a pretty good toolbelt of intuition to get started with cosets in group theory. 
A: The multiples of $5$ form a group (for addition), the numbers that are a multiple of $5$ plus one, like $6, 11, 16, \ldots$ and so on, form a coset of the group. If you add two to a multiple of $5$ like $7, 12, 17, \ldots$ and so on you have another coset. Check that there are five cosets in all (the group itself counted as a coset). Now the nice thing about these cosets is that you can add and mulitply them. Take any number from the first coset ($6, 11, \ldots$) say $11$ (such an element is called a representative of the coset), and a representative of the second coset, say $17$, and add them up giving $28$. Now $28$ is a multiple of $5$ (i.e. $25$) plus three. Check that we obtain the same result if we had chosen other representatives. Do the same with multiplication. If you understood this you have discovered modular arithmetic and you made aquaintence with the ring $\mathbb{Z}_5,+,.$
