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.
I think the use of homomorphisms is to identify one algebraic structure with another. With that identification we can say that two algebraic structures are same structurally i.e they are same when we view them with respect to algebraic operations involved in them although they may seem different with open eye.