Definition of an Algebraic Objects How did the definition of Algebraic objects like group, ring and field come up? When groups were first introduced, were they given the 4 axioms as we give now. And what made Mathematicians to think of something like this.
 A: I'm certainly no historian, so let me give my impression of the history in a CW answer and others can improve on this answer.
Groups were understood as permutations of objects long before they were understood as abstract groups, just as manifolds were understood as nice subsets of $\mathbb{R}^n$ long before they were understood as abstract manifolds.  Most notably, Galois worked with permutations of the roots of a polynomial in order to understand the solvability of polynomials by radicals.  Similarly, number theorists like Kummer worked with concrete examples of number rings and fields.
Some of the people responsible for the spread of the abstract point of view are (in no particular order) Hilbert, Noether, Artin, van der Waerden, (Dedekind?), .... together they more or less determined the abstract algebra curriculum as we now understand it.  Their reasons for doing everything axiomatically include that the proofs are shorter and more powerful this way; in particular, algebraic geometry needed rigorous tools and commutative algebra provided them (and Zariski used them).  
One big payoff of the abstract point of view is that people now recognized fundamental groups and homology groups in topology as groups.  Before the abstract study of groups these objects, while defined in some sense, weren't recognized as groups because they didn't permute anything.  
A: Let me address the question: "what made Mathematicians ... think of something like this"?
The answer is: Galois, in studying the problem of factorization of polynomials, realized that
reasoning about the symmetries of the roots could be more powerful a tool than studying explicit formulas (which, loosely speaking, had been the basic method in the theory of equations up to that time).  
As this structural/conceptual point of view began to reveal its power in solving difficult concrete problems, more and more mathematicians began to think in this way.  The structural concepts were then isolated from their concrete settings, and this is how they are taught today.  But the motivation was, and for many remains, the applications of these abstract notions to concrete problems.  (A standard but helpful example is the pivotal role that group theory, Galois theory, cohomology, and ring theory played in the proof of Fermat's Last Theorem.)  
Furthermore, nowadays we train ourselves and our students to recognize any hint of structure in a problem, and to exploit such structure to the hilt.  So these concepts have become basic tools in the problem-solving toolkits of contemporary mathematicians.
A: See the books A History of Abstract Algebra and Episodes in the History of Modern Algebra (1800-1950).
A: With respect to the introduction of fields, Erich Reck in his "Dedekind's Contributions to the Foundations of Mathematics", cites Dedekind as being the first to introduce this concept into mathematics, in the context of his work in algebraic number theory. Reck's article goes into some detail about Dedekind's other contributions to mathematics that seem like they may be of interest, for example, his work on lattices.
