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 :)
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.