I am new to the field of Abstract Algebra and so far it's looking to me quite tough. So far I have encountered the following books in group theory - Contemporary abstract algebra by Joseph Gallian and Algebra by Michael Artin. But can someone suggest me a book which has theorems and corollaries explained using examples and not just mere proofs?
Perhaps A Book of Abstract Algebra by Charles Pinter? It's very cheap, and if I remember correctly, provides motivation for the theorems/corollaries/etc. It's a small book, and there's a lot of stuff it doesn't cover, but I think it would prepare the reader well for more advanced treatments of algebra.
"A first course in Abstract Algebra" by Fraleigh is a great introduction to Algebra and it is packed with examples. My experience with this book is I learned Algebra on my own out of the first half of this book and took a course out of the second half. It is also available for free online if cost is a factor.
In addition to great recommendations such as Fraleigh and Pinter, you might want to try Visual Group Theory by Nathan Carter. He uses numerous examples and illustrations, with many Cayley diagrams. The exposition is gentle and doesn't reach the group axioms until Chapter 4 (out of 10), yet still manages to give a taste of Galois theory by the final chapter.
I seriously can't believe nobody has mentioned this yet, but Dummit and Foote's "Abstract Algebra" 3rd edition is perfect for those starting out, and will serve as a great reference later on down the line as well.
They introduce an abstract concept, then immediately follow it up with at least 3 different examples to give you a good sense of the objects you're dealing with.
On top of this, the examples they give are highly relevant to what you will see down the line in your other classes.
You seemed to be new to abstract algebra and you just want to feel the subject, I mean, to see how the results work and why they are how they are instead of just looking at proofs. Maybe my recommendation could sound a little controversial but here it goes:
1) First you need some motivation and a book which explains each thing in easy terms and easy words. What it is each concept without playing to much attention to proofs, just to get the intuition. I would recommend Visual Group Theory by Nathan Carter.
2) The second stage is an easy to follow book with a lot of details in the proofs and with examples. At the same time it should not be very big since you must read the proofs here. I recommend A Course in Group Theory by J. F. Humphreys. It has solutions to the exercises. It would be much more than enough for your undergraduate Group Theory. (I would cover until Sylow Theorems and then study some ring theory)
3) You can combine 2) with Topics in Group Theory (Springer Undergraduate Mathematics Series) by Olga Tabachnikova. (this is optional)
4) Complement 2) with the excellent expository papers on Group Theory and in concrete in group actions written by K. Conrad: http://www.math.uconn.edu/~kconrad/blurbs/ This is a must.
5) Feel free to jump to Ring Theory. If you have work over the previous material you should be prepared to take the book you want: Abstract Algebra by Dummit and Foote is famous. To be controversial I would suggest: Mahima Ranjan Adhikari, Avishek Adhikari-Basic Modern Algebra with Applications, Springer India (2014). This is a book which covers some interesting topics such as a lot on exact sequences (useful for Algebraic Topology), and lots of applications of Abstract Algebra to make the theory a bit less dry.
Hope this helps!
Rotman's "Introduction to the Theory of Groups" is a great book. It is focused on groups only (unlike some books on abstract algebra that sometimes skim over the subject), and Rotman's style makes it very readable. He usually includes proofs to every claim, a good deal of examples, and useful exercises.
I want to include "Abstract Algebra" by I. N. Herstein. Admittedly, I chose this gem based on its reviews, but the author offers a unique writing style, extremely lucid explanations and tries to motivate topics using historical developments in the field, which helps immensely to clarify seemingly unintuitive topics.
I understand there is another well known book called "Topics in Algebra" by the same author. It is described as
"... a classic and influential [book] ... which dominated the field for 20 years."
I recommend Abstract Algebra Structure and Application by Finston, Applied Abstract Algebra by Lidl and Pilz, and Abstract Algebra Theory and Applications by Judson. While these books may not have quite the expository depth of the more theoretical titles, they provide a gentle introduction to Algebra firmly grounded in practicality rather than theory. Each book contains numerous applications to areas of computer science and cryptography, and when paired with a book like Artin's can provide a strong introduction to the subject.
I am quite suprised nobody has mentioned this book so far. I suggest you try "Algebra" by Michael Artin. This book is suitable for a beginner to intermediate level course in abstract algebra, especially group theory and uses the linear group as one of its recurring examples, which I.N.Herstein's book doesnt
Less (and more) than what you're looking for, but very interesting: Abstract Algebra done Concretely.
As Brandon Thomas said in his answer I utterly recommend Dummit Foote's "Abstract Algebra". It is a classical introductory book with a highly comprehensive section on group theory.
I also recommend "A course on group theory" by John Rose. It is a cheap, complete and easy-read book, and It includes some of the most important (even still open) problems on group theory.
Everybody here mentions mainly books generally about algebra, they are all great and might have good expositions on groups. But let me add some books specifically on group theory.
i) Derek Robinson, A course in the theory of groups
Well-written and touches on many topics like free groups, transfer, representation theory, cohomology, permutation groups and so on; including some I have not seen in other introductory books.
ii) M. Hall, The theory of groups
A classic. Would be a good addition to Robinson's book, as the focus is slightly different and it also has a distinctive section on group theory and projective planes.
iii) John Dixon & Brian Mortimer, Permutation groups
Focuses on groups acting on some sets, which could be regarded as subgroups of the symmetric group. If you want to understand group theory it might be a good idea to read specifically more about permutation groups and groups acting on some set, as this is one natural setting where groups arise.
iv) John Dixon, Problems in group theory
This is a selection of exercises with solutions from many topics ranging from basics, to permutation groups up to character theory. It is published by Dover, so it isn't that expensive and might be a good addition.
Also there are other books, but I just heard about them and had not read them yet: I always heard that M. Suzuki's books Group Theory I & II are extremely well-written, but I haven't taken a look, but maybe you want to check them out. Also for permutation groups, H. Wielandt's Finite Permutation Groups and P. Cameron Permutation groups are often mentioned.
And also a book I have looked in from time to time is Antonio Machi: Groups: An introduction to ideas and methods, this is for the beginning student and has many exercises with solutions in it. I guess this might fit well what you are asking for.