Okay:
Flatwounds: These strings are smooth as you run your finger up and down them. They are similar to cello/violin/viola/double bass (sometimes) strings. They typically are of higher tension then roundwound strings. They usually give a deeper tone. Famous users of flats on a P are James Jamerson, Steve Harris (however, his sound is not like most percisions with flats), Phil Lynott, Sting, John Deacon, and more jazz players than I can ever remember. They get deeper sounding with age, and if you are using flats, that is probably something you want.
Roundwounds: These are made like guitar or piano strings. They are rough to the touch and give a brighter and more defined sound. The two most common types of roundwound are Nickle and Stanless Steel. Stainless Steel strings are a bit harder on the fingers and have more high end (at the expense of a shorter life) while nickles are easier on the fingers and last longer at the expense of high end. Off the top of my head, I can not think of any famous recordings that were made with percisions running rounds, but I am sure they exist.
Which is best for you? Depends. Do you like a bright sound, low tension, and/or do a lot of slap/pop? Then go for roundwounds (I personally like Rotosound SwingBass 66 Stainless Steels, but try as many as you want before finding your favorite).
Do you like a deeper sound, high tension and never want to change your strings again? Go flatwound (My favorite flats ATM are D'Addario Chromes 45-105s but if you ask anyone on this board, they will probably say TI Jazz flats. I have not tried those yet as they are expensive.)
Also, there are types of strings that seek to combine the best of flats and rounds. These are called either pressurewound or groundwound. They consist of a roundwound string that has had all the edges polished off. I do not have enough expertise on these to give advice.
Sounds complicated? Don't worry. It all boils down to these three questions:
Who are some of your favorite bassists? Who are some of your influences? What are some of your favorite bass tones?
From there, the good people on this board will be able to tell you what strings to get.
Hope this was helpful!
Charles F. Sporn