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!
I have always understood a postdoc to be a "post doctoral" position, that is a paid post. The idea of a postdoc position is to do research roughly in the area of the PhD, with the person gaining experience that would possibly be of help in gaining a full time position.