Abstract algebra is usually introduced either to senior undergraduates or first-year graduate students in mathematics. (Disclaimer: this is in the U.S. -- I can't speak for other countries' systems.) It is called "abstract" to distinguish it from the algebra learned in high school. Mathematicians generally understand the word "algebra" to refer to abstract algebra, and will say "high school algebra" or some such when they need to refer to the more basic subject.
There is no exact definition of what constitutes abstract algebra, since the various fields of study in mathematics overlap and intertwine quite thoroughly. One definition is that algebra becomes abstract when we allow variables to represent objects other than just numbers, such as polynomials and matrices. Abstract algebra even allows variables to stand for operations. Thus, when an expression such as "a*b" appears in abstract algebra, a and b are most likely not numbers, and * is most likely not multiplication.
Another definition, which is a bit more rigorous:
Algebra is the study of arbitrary operations and relations on arbitrary sets.
Another way to look at abstract algebra is that it involves removing key properties and concepts of basic mathematics from their context so that they can be studied in isolation from one another. The areas of "basic mathematics" include counting, arithmetic, symmetry, order, and combinatorics, among others.
For example, the set of all integers is closed under the operation of addition. (That means that, if m and n are integers, then so is m + n.) Addition is commutative and associative. There is an element, 0, such that n + 0 = n for every integer n. Also, every integer n has an inverse, -n, such that n + (-n) = 0. If you make up a set and an operation on the set that have all these properties, you have an algebraic structure called an Abelian Group.
Groups, rings and fields are generally the first objects studied in abstract algebra courses. Beyond these basics, there are a great many types of algebraic structures -- modules, semigroups, monads, lattices, Lie algebras, Boolean algebras, and innumerable others.
Algebra is a fundamental part of advanced mathematics. Most other fields make use of algebraic methods in one way or another. Fields such as algebraic geometry, algebraic topology, and algebraic logic are important areas in modern mathematical research. | 
