Who actually is a Post-Doctoral fellow?
There are many formal roles that generally fall under the category of "post-doc": 


The simplest is as a post-Ph.D researcher working with a faculty mentor and doing their own research
Some postdocs have a role as "lab manager": they help with advising students.
In addition, if given an appropriate title, a postdoc (as "visiting/research faculty") can write their own grants or collaborate on grants that might support them.
The limit of the above is a pure soft-money position that is not "supervised" by a faculty member. Such faculty are also often called 'research faculty', and may be many years away from a Ph.D
In wetlabs, a postdoc role might also be as a lab technician or lab manager, handling supplies, tech work (making knockout mice for example).