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: 
Groups, rings (especially fields) and many other objects in mathematics form so called categories. I.e. the collection of all rings (whatever this is) forms a category. To 'compare' the objects in a category, one needs morphisms between them. (This is not the only need for morphisms, but one of them). To say that one object is 'bigger' or that two objects are isomorphic, one needs the notion of morphism.

Morphisms are generally maps (though not all morphisms need to be maps) that respect the structure of the objects. 

Of course, morphisms have other useful applications. Matrices for example (i.e. morphisms of vector spaces [or modules]) are connected to linear equations and so on...

Morphisms play very many very important roles in mathematics :)
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.