I could explain that to you, or just link you to the pretty new wikipage we made for that.
I'll do the explaining first.
The addition I have to my name ("The Listmaker" ) is a custom rank. Only few PEELers have that and they earned it by doing special things, mettingthe admins, or behaving in a certain funny way here.
For more on that read the wikiPEELia article on Custom Rank
Asking to get such a rank will in all cases lead to the fact that you'll never get one.
The official ranks, like Benders_Fan posted above, are achieved by posting. Every one here fits into one of these ranks. I.e. I am an "Urban Legend" because I have made more than 3001 but less than 6667 posts.
From Starship captain on, the action of going up in the rank becomes special for most PEELers. We announce that we went up in ranks by naming special things in special threads.
For Starship Captain we name a ship we would own and add it to a fleet.
For Professor we name what we are teaching, or what field we are a professor of.
For Liquid Emperor we name who was last Liquid Emperor of PEEL and drink him.
For Urban Legend we name what kind of a legend we are.
For SpacePope we name our religion.
the wikiPEELia article for ranks is here