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: 
Homomorphisms do a lot of things:

1. They can be used to show two structures are identical (isomorphism)
2. They can be used to show one structure is a substructure of another (monomorphism)
3. They can be used to show one structure is a quotient of another (epimorphism)
4. General homomorphisms are kind of a mixture of the above, carrying a certain amount of data about the relationship between the domain and codomain.

This list is not exhaustive, it's just meant to sample some of what homomorphisms do.
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.