Let $(A,\triangle)$ and $(B,\triangledown)$ be groups. $f:A\rightarrow B$ is a homomorphism if $\forall x,y\in A$
$ f(x\triangle y) = f(x)\triangledown f(y) $
This is basically saying that $f$ preserves the group structure when it maps element of $A$ to elements of $B$. Homomorphisms tell us about the similarities between two groups.
For example, take the integers under addition modulo 2 and 4, $\mathbb Z_2$ and $\mathbb Z_4$. Let $f:\mathbb Z_2 \rightarrow \{0,2\} \subset \mathbb Z_4$ such that $f(0)=0$ and $f(1) = 2$. $f$ is a homomorphism (note that it is injective). Since the two groups are abelian it suffices to show that $\begin{align} &f(0+_20) = f(0)+_4f(0) = 0+_40 = 0 = f(0) \\ &f(0+_21) = f(0)+_4f(1) = 0+_42 = 2 = f(1) \\ &f(1+_21) = f(1)+_4f(1) = 2+_42 = 0 = f(0) \end{align}$
This tells us that the structure $\mathbb Z_2$ is equivalent to the structure of the subgroup $\{0,2\}$ of $\mathbb Z_4$.
It is important to note that homomorphisms fix the identity element. If $e$ is the identity element in $A$ and $x \in A$ then $f(x\triangle e) = f(x)\triangledown f(e)$ and $f(x\triangle e) = f(x)$ (simplifying inside of $f$). So, $f(e)$ must be the identity element for $B$.
An isomorphism is a bijective homomorphism. If two groups are isomorphic they have the same group structure.
An epimorphism is a surjective homomorphism. If there is an epimorphism from $A$ to $B$, it implies that $B$ is isomorphic to some quotient group of $A$.
A monomorphism is an injective homomorphism. A monomorphism from $A$ to $B$ implies that $A$ is isomorphic to some subgroup of $B$ (as we saw above).