TITLE: What is Abstract Algebra essentially?
QUESTION [45 upvotes]: In the most basic sense, what is abstract algebra about?
Wolfram Mathworld has the following definition: "Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic structures rather than the usual number systems. The most important of these structures are groups, rings, and fields."
I find this, however, to say the least, not very informative. What do they mean by abstract algebraic structures? Along these lines, what are groups, rings, and fields then?
I've been told by a friend that groups, essentially, are sets of objects, although, this still leaves me wondering what he means by objects (explicitly).
I don't need anything rigorous. Just some intuitive definitions to give me some direction.
Thanks!

REPLY [2 votes]: Abstract algebra is an area of pure mathematics; see lists of mathematics topics. There is overlap between different areas of math, but if one had to divide math into two areas, it would have to be algebra and geometry.
Geometry is the study of shapes, graphs and points in space. If you are working and studying in this area, you would almost certainly be drawing interesting pictures.
If you are exploring abstract algebra, you would have far fewer pictures in your work notes. You would be manipulating structured expressions, strings and words.
David Hilbert's quote certainly applies if you are immersed in abstract algebra:

Mathematics is a game played according to certain simple rules with
  meaningless marks on paper.

Finally, for another perspective, see connecting Algebra with Geometry.