Question: In group theory we have encountered the concepts of homomorphisms as maps from one group to another, and the same thing with ring and field theory and we ave called them respectively homomorphisms of groups (of rings, and of fields). Also in linear algebra we have linear maps.
So my question is this:
What is the role of such maps in this algebraic structures?

Answer: 
Another function of morphisms is as a way of taking an unfamiliar or apparently complex algebraic object, and placing it in a more familiar context, or one where there are known analytical tools and methods - as in representation theory.

The point of view you take depends rather on whether you are more interested in studying particular objects, or alternatively classes of objects.
Let's suppose for the moment that you really only care about isomorphisms. That is, you really only care about telling whether two groups are isomorphic, or something. Great! But how do you write down isomorphisms? It's hard. Often you will be able to write down a map and you'll want to show it's an isomorphism. *Before* you show it's an isomorphism, what is it?

It's probably a homomorphism!

So even if you only care about isomorphisms, in the course of proving things about isomorphisms you will very likely end up having to talk about homomorphisms which are not necessarily isomorphisms (at least you don't know that they are), so you might as well talk about them from the outset. Moreover, 

* you can use homomorphisms to construct isomorphisms,
* even homomorphisms which are not isomorphisms can be used to prove useful statements, e.g. that there do *not* exist certain isomorphisms,
* etc.

But really to understand this you should do a lot of mathematics. It will become clearer to you as you gain experience with working with maps.