Abstract Algebra generalizes the types of things you can "do algebra" on. For example, in group theory, you often work with permutations instead of numbers. Things start to look more like the algebra you're already familiar with when you get to ring theory and field theory, but there are still plenty of unusual objects in those areas.
Essentially what you do is take away as many previously held assumptions as you can and try to reduce algebraic problems to only what is actually needed. In this sense, you seek the broadest view possible, and in doing so are able to apply what you learn to a much larger range of structures.
Modern cryptography is entirely built off of abstract algebra, in particular from the theory of finite fields. Coding theory is also very algebraic, with both direct applications and "how to think" applications.
Group theory is used in quantum mechanics, in fact there is arguably an entire subdiscipline of algebra devoted to this.
Other areas of math require an appreciable understanding of abstract algebra when moving to advanced levels. Obviously anything with "algebraic" in the title is going to need it (e.g. algebraic geometry, algebraic number theory, etc.) Other examples, topology uses fundamental groups, and functional analysis uses tons of abstract linear algebra. For this reason people often say that learning abstract algebra is learning the language of higher mathematics, although some (myself included) find the field interesting for its own sake.
As for when it is right to start studying abstract algebra, you are ready as soon as you are able to write proofs. If you are comfortable with direct proofs, contrapositive/contradiction, and at least a little induction, then go for it. If not, buy this book.
If you think you're ready, my personal preference has been to study with several books at once. Many will recommend Dummit and Foote, which is a great book, but in my opinion may be too heavy for a beginner. I started on Fraleigh, which was okay. (If you are self-studying, Fraleigh may be good because there is a solutions manual available online.) Although I have not used it myself I have heard that Artin is heavily motivated by examples (very important for abstract algebra) and thus a good place to start. I don't know too much about online abstract algebra resources, but in general I think books are better. So I recommend you procure one of those, either from the library, through the back alleys of the internet, or with actual money if you have it.