if you think of a group as being like a big "blob", made out of its elements (particles), then a subgroup is like a "chunk" containing containing the identity element (which lies in the very middle of the blob). the idea is, can we make a smaller group out of bigger chunks than the particles?
and those chunks are what cosets are. we name the chunks by picking out a certain element, and calling it a representative of that coset. but this is somewhat arbitrary, a coset is a chunk, and might have several equivalent names.
now our blob isn't amorphous, it has an internal structure (like crystals do), a certain regularity imposed on it by the fact that we have a multiplication. we can only make chunks that are the same size as a subgroup.
abelian groups are the nicest, because we can go back and forth without paying attention to which "direction" came first (multiplication is like saying: start at point a, and then use point b as an instruction to travel to point ab). in fact, abelian groups are "almost" like vector spaces, they're just not "big enough" usually to have the full force of geometry.
for example, the integers, can be thought of as lying on a line. what happens when we "mod out nZ (the multiples of n)"? well, we make all the multiples of n indistinguishable. so all these "dots" (spaced every n dots apart), all contract to a single point (the "origin", or "0").
and instead of a line, now we have dots arranged in a circle, with n stacked on top of 0, n+1 stacked on top of 1, n+2 stacked on top of 2, and so forth (the line "wraps around the circle" infinitlely many times, so each "dot" is now a coset composed of an infinite number of dots). so it's no longer important what each number sitting on a dot used to be, the only thing that matters now is "how far from 0 it is".
we actually have something that acts like this: a clock. for this reason, integers mod n, are sometimes called "clock numbers". the circular nature of this group gives rise to the name cyclic.
now, with a general group, G, and a subgroup H, we try to mimic this behavior of the integers mod n, as much as we can. we want the cosets, or translates of H by g, gH, to act like k + nZ (move the 0-dot (clump) by k). moreover, it would be nice if the "clumps" had a group structure (thus replacing a complicated group structure with a simpler one, which might be easier to deal with).
so we would like (xH)(yH) = xyH. that would make everything nice and simple for us.
well, let's see what goes wrong:
we pick an element xh in xH, and an element yh' in yH, and multiply them:
xhyh' = xy...? hmm, the trouble is, that hy might not be equal to anything like yh". the left coset of H that contains y might not be the same as the right coset that contains y. this is a problem, it stops us dead in our tracks. and since there are LOTS of non-abelian groups, this is bound to come up over and over again.
what to do? well, if H is a subgroup that "kinda commutes" with everything, in the sense that yH = Hy for any y in G, that will fix everything. then hy = yh", and we can continue:
xhyh' = x(hy)h' = x(yh")h' = xy(h"h')...success! in coset form:
xHyH = x(Hy)H = x(yH)H = xy(HH) = xyH (since for any subgroup, HH = H).
personally, i think "quotient group" is a bad name, because nothing like "division" is going on. so what G/H really means, is G, in H-sized bites. and (mostly because we have non-abelian groups, which are the troublemakers of group-land), H has to be a special kind of group, called normal, or else the "clumps" don't behave well.