This is just my opinion and how I think about it.
Your question is not a bad question, but it is difficult to answer because it is going to depend on peoples personal feelings. One might "divide" math into areas in one way, while another might divide another way. Besides that, there is the question about how deep you want to make your division - i.e. how many areas are we talking about. Then there is the "problem" that with pretty much any division there are going to be overlap. So if you go to conferences, you might see a person going to different conferences in different areas because he/she works in an area that that overlaps two other areas. So it is really diffictult.
However, when I try to explain what I do, I usually start out by diving math into the areas
- Applied math
(Yes, some people like to put the applied math under analysis).
Now you might ask: "So what do these 'different' areas each cover?" or "What distinguishes one area from another?". This is even harder to give a good non-offensive answers to, so I wouldn't even try.