In math, we normally proceed by learning elementary arithmetic and then elementary algebra. For me algebra is all about assigning a symbol to an unknown value and manipulating it to find its value.
Now I don't understand how concepts like groups, fields, rings etc. fall into the algebra category.

Wikipedia tells about the history of abstract algebra in brief:

I've not studied general polynomial equations and higher degree forms of diophantine equations so I don't understand this.

Can someone please explain me how concepts like groups, fields, rings etc. fit into my understanding of algebra which is basically assigning symbols to unknown and playing with it.

Without giving you a full course in college algebra, algebra is a little more involved than "assigning a symbol to an unknown value and manipulating it to find its value."

Why should you be able to do this at all?

Algebra shows us that if we, for instance, take a real number and multiply it by another real number, we should get a real number result. Or, that the order in which we multiply two real numbers together is immaterial to getting the product, e.g. A × B = B × A. But what if the order in which two numbers were multiplied together made a difference in the value of the product obtained, so that A × B ≠ B × A? Abstract algebra helps to generalize the ramifications on other aspects of arithmetic what happens in such situations.

Algebra studies the solutions to polynomial equations with real number coefficients and positive integer powers. You should know how to solve for the unknown value in a simple linear equation, and you should have learned the quadratic formula for solving second degree equations. There are complicated formulas for solving cubic and fourth order equations, but no such formulas have ever been found which solve fifth and higher order equations. Why is that?

Well, a very young Frenchman named Galois pondered this question and showed that it was impossible for a formula involving a finite number of arithmetical operations to be developed to solve fifth order equations and higher degree equations. He did this by showing that the different solutions to a given type of equation could be organized into what is known as a group, and that only certain types of equations had solutions which fell into what are now called "solvable groups".

Don't worry if you don't understand this. Some of the best mathematical minds of Galois' generation and later had much difficulty grasping what Galois had done, primarily because Galois was working with math concepts which were so unfamiliar at the time.