What is Abstract Algebra essentially? 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!

Mathematics has to do with sets. There is no definition about what the set is, but we all know that set is made up by elements. Many problems in nature can be represented by sets, and the relations of the elements on these sets. The mathematical discipline that studies THE RELATIONS OF ELEMENTS on a given set is called algebra. There are many good properties of relations that a set can have, and from those properties we classify classes of 'sets with their relations' (algebraic structures) and call them as groups, rings ect.
