Question: I am a PhD student in Mathematics. I wanted to know what exactly is the difference between PhD and Postdoc. Is it just the research that one does after PhD(kind of second PhD?)?

Answer: 
I want to address this question by citing my current Master's advisor when I asked him the same question:

1. In the Bachelor's, you learn the basics of the area.
2. In the Master's, you learn what people are doing in the field and how to read scientific papers.
3. As a PhD, you learn how to develop your own research and start having new ideas on the field.
4. As a Post-Doc, you learn how to mentor other people's research while producing your own work without supervision.
5. When you reach Professorship you only have to look smart!
The first and foremost difference between the two is that a PhD is "awarded" after defending a thesis (plus additional duties depending on the department). On the other hand, a PostDoc is a temporary working position that is assigned by some institution, whose completion does not require any defence.

What people do or do not do, researchwise, in their PhD and PostDocs is very much dependent on the area and on the department (you may look at other answers that go more in details on this and that I do not want to overwrite, as I would just copy and paste them).