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: 
One of my professors once said this about ring theory:

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> To study rings, you have two choices. You can take a particular ring, and stare at it intently for a long period of time until you discover interesting things about that ring. Then you write them down.
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> Or you can study how the rings "acts" on other structures, by studying the collection of all modules on that ring, and thus deduce interesting things about the ring. It turns out that the latter method is more fruitful, easier to generalize, and allows one to go from one ring to another ring more easily.
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Studying the modules of a ring can be seen as a special case of studying homomorphisms (an $R$-module structure on an abelian group $A$ is equivalent to a ring homomorphism from $R$ to the endomorphisms of $A$, which are a ring under pointwise addition and composition. (Which is why I mention it)

The example of vector spaces is very apt. Vector spaces are nice objects, no doubt; but if all you do is stare at a particular vector space, you can say some interesting things (dimensions, subspaces, and so on), but it's not until you bring in linear transformations that the full power of vector spaces really becomes apparent. It is through linear transformations that we can actually *do* things with vector spaces: use them to solve systems of equations (linear and differential), to study Markov systems, to solve least squares problems (which amount to constructing certain kinds of projections, which are types of linear transformations), etc.

There is a whole philosophy of mathematics that says that the best way to study objects is to study the maps that "respect the structure", and not only for algebraic objects: topology is not just the study of topological spaces, it's the study of topological spaces *and continuous maps between them*. Analysis is not just the study of the real numbers, it is the study of the real numbers and *real valued functions*. Differential geometry concerns itself with differentiable maps. Etc. Turns out that maps give you a fruitful way of studying the structure.

In algebra, an important role is played by "congruences", which are equivalence relations on the underlying set that are compatible with the operations. For example, in the ring of integers, the usual modular congruences, $a\equiv b\pmod{n}$, give an equivalence relation that "respects" the ring structure on $\mathbb{Z}$, in the sense that if $a\equiv b\pmod{n}$ and $x\equiv y\pmod{n}$, then $a+x\equiv b+y\pmod{n}$ and $ax\equiv by\pmod{n}$ (the sum of classes is the same as the class of the sum; the product of classes is the same as the class of the product). Congruences correspnd, via the isomorphism theorems, to surjective morphisms. 

Morphisms let you take one particular structure and study it from other points of view, in other contexts; it allows you (via "quotients") to focus on particular details while ignoring other part of the structure that may be "in the way" (e.g., the study of "parity" is made simpler by considering the quotient $\mathbb{Z}/2\mathbb{Z}$, rather than $\mathbb{Z}$ itself). 

Morphisms let you understand inherent symmetry in an object; Klein proposed studying geometry almost *exclusively* in terms of certain kinds of symmetries and certain kinds of morphisms (projective transformations). Galois theory is founded on the idea of understanding the "symmetries" that may exist between different roots of the same polynomial (in the form of automorphisms).

All in all, morphisms are just a very rich way of studying objects. You can think of them as a way of letting you poke, examine, cut open, x-ray, fold, and generally manipulate your objects of interest so that you can get a better sense, not only of what they are, but how they interact with other objects and other contexts.
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.