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.