BOOKS:

Textbook:
• "Fundamentals of Abstract Algebra" by D. S. Malik, J. N. Mordeson, M. K. Sen, McGraw-Hill, 1997.

  
  Supplementary Books:

• "Contemporary Abstract Algebra" by J. A. Gallian, D. C. Heath and Company, 1994.
• "Algebra: Abstract and Concrete" by F. M. Goodman, Prentice Hall, 1998.
• "A First Course in Abstract Algebra" by J. J. Rotman, Prentice Hall, 2000.
• "Abstract Algebra" by T. Shifrin, Prentice Hall, 1996.
• "Algebra" by M. Artin, Prentice Hall, 1991.

  
  If these books are difficult for you, then before attacking them see:

• "A First Course in Abstract Algebra" by J. B. Fraleigh, Addison-Wesley.

  
  Graduate (hence more difficult!) Algebra Texts:

• "Abstract Algebra" by Dummit and Foote, Prentice Hall, 1991.
• "Algebra" by T. W. Hungerford, Springer, 1974.
• "Algebra, a Graduate Course" by I. M. Isaacs, Brooks/Cole Publishing Company, 1994.
• "Algebra" by S. Lang, Addison-Wesley, 1993.

  
  More About Groups:

• "Fundamentals of the Theory of Groups" by M. I. Kargapolov and J. I. Merzljakov, Springer.
• "A Course in the Theory of Groups" by D. Robinson, Springer, 1996.

  
  More About How Algebra and Geometry can be Combined:

• "Glimpses of Algebra and Geometry" by G. Toth, Springer, 1998.
• "Finite Reflection Groups" by Benson and Grove, Springer.
• "Groups and Symmetry" by M. A. Armstrong, Springer, 1988.

  
  More About Rings:

• "Rings, Modules and Linear Algebra" by B. Hartley and T. O. Hawkes, Chapman and Hall, 1970.
• "Introductory Lectures on Rings and Modules" by J. A. Beachy, Cambridge University Press, 1999.

  
  More About Fields:

• "Field Theory" by S. Roman, Springer.

  
  If you are looking for challenging problems at the undergraduate level, then see the exercises in:

• "Topics in Algebra" by I. N. Herstein, Wiley and Sons.

  
  What about some history?

• "A History of Mathematics" by Carl B. Boyer.
• "Mathematical Thought from Ancient to Modern Times" by M. Kline.
• "A Concise History of Mathematics" by D. J. Struik.