Don't confuse this with the notions of *hom*eo*morphisms* and *homotopy*, which are used to express ideas like your coffee mug being topologically equivalent to a donut. Those aren't notions we'll be talking about. *Homology* is yet another notion with a similar-sounding name---also with connections to topology---that is different from what we're talking about here.

As with isomorphisms, let's still require a (homo)morphism to be a function from one universe *onto* the other (sometimes this isn't required).
And it still has to be the case that $f(x\star y) = f(x)\bullet f(y)$ and so on.
But we will lift the restriction that the translation function be injective.
For example, consider the groups $\Z_6 = \tuple{\set{0,1,2,3,4,5}, + \bmod 6, 0}$ and $\Z_3 = \tuple{\set{0,1,2}, + \bmod 3, 0}$, and the function $f$ defined as follows:
> $f(0) = 0$
> $f(1) = 1$
> $f(2) = 2$
> $f(3) = 0$
> $f(4) = 1$
> $f(5) = 2$
This function $f$ is a (homo)morphism between our two groups. It's not an isomorphism because we've collapsed some distinct elements of the first group into single elements of the second group.
Of course, the fact that *this* function $f$ is not an isomorphism between the groups is compatible with there being some *other* function which is. However, as a matter of fact no isomorphism exists in this case and these two groups are merely homomorphic.
A (homo)morphism gives us a way of retaining *some* structure from the original algebra, but perhaps disregarding other parts of its structure.
### Endomorphisms and Automorphisms ###
Sometimes (homo)morphisms are functions from one algebra onto *that very same algebra*. This doesn't necessarily mean they are bijections. If the universe of the source algebra is infinite, it can be mapped onto itself while at the same time collapsing several elements from the source into a single element in the target. Here's an example of this; it's reminiscent of the homomorphism from $\Z_6$ to $\Z_3$ described above.
Consider all the rational numbers $0\le q \lt 1$. Let that be the universe of a group. The operation on our group is addition mod $1$. That is, $\frac12 \star \frac34 = \frac14$, and so on. Now we are going to describe a (homo)morphism from the group onto itself. It maps every element of our group to twice its value---except we wrap around when we get values $\ge 1$. So, for example, $\frac14$ gets mapped to $\frac12$, but so too does $\frac34$. Given how dense our group's universe was to start with, every value in the "target" will get at least one (and indeed, will get exactly two) values from the "source" mapped onto it. The group we end with is the very same group we started with: the rationals from $0$ up to (but not including) $1$, with the operation of addition mod $1$. So this is a (homo)morphism from a group onto itself, where the translation is onto but not injective.
(Homo)morphisms from an algebra onto itself, whether bijections or not, are called **endomorphisms**. I don't expect you to remember their name.
When we have a endomorphism that *is* bijective---that is, an *iso*morphism from an algebra onto that very same algebra---this is called an **automorphism**.
Every algebra has a "trivial" automorphism, which is just the identity function on its elements. In some cases, there are also other more interesting automorphisms from the algebra onto itself. These are easier to illustrate than endomorphisms which aren't isomorphisms. For example, consider the isomorphism that maps $\Z_4$ onto itself, where:
> $f(0) = 0$
> $f(1) = 3$
> $f(2) = 2$
> $f(3) = 1$