Pros:
Doing more research (i.e. 2nd piece of major research if you haven't done research before PhD)
Become more knowledgeable/specialised in your research area (some suggest postdoc should be in a different area to PhD but can be within same field or you can switch to a related field if you can apply your knowledge/techniques)
Good research training for academia leading to more publications
Raises your research profile - lets you become more established. Most do a postdoc(s) before getting a lectureship (in sciences anyway, it's possible to go to a lectureship from PhD in social science)
Cons:
Short-term position
Very hard to get funding
If you don't want to go to academia it is a lot of hassle trying to get one - e.g. you can apply for advertised positions or approach a potential supervisor who will co-write a proposal with you to apply for funding.
Most people only pursue the postdoc route if they like research and want to stay in academia. It's not the easiest position to find especially in current economic situation! The funding often seems like a lottery as only a small percentage of proposals get funded by research councils, other funding bodies and charities in comparison to the number of proposals received. Research councils often have themes they promote so a lot of the funding will go to those projects.