Coffee was introduced to Europe and England by traders returning from Arabia. By the early 1600s, there were coffee houses in London where the coffee was made by adding finely ground coffee to a pot of boiling water. Ever had "Turkish coffee"? Well, that's exactly it. Once the ground coffee had settled to the bottom of the pot, the coffee was poured off into cups. Just like with Turkish coffee today, there was always a residue of coffee grounds in the cup. To prevent this outcome, the "coffee sock" became popular which was quite literally an old sock in which the coffee grinds were wrapped and then added to the pot to prevent the ground coffee ending up in your cup. The one unwanted outcome of this, however, was that the taste of the coffee was typically tainted with the flavor of the fiber, be it burlap, wool, or cotton. (No, not the taste of smelly feet; let's give them the benefit of the doubt that the sock was washed before being put into service!)
Credit for the modern-day paper filter goes to a German woman called Melitta Bentz who patented a paper coffee filter in 1908 after experimenting with various paper types and finally settled on her son's blotting paper used for keeping his handwriting from smudging. (Yes, I'm old enough to remember and have used blotting paper as a kid at school, though I'm sure many younger people reading this will have had no experience of using blotting paper). You can still see Melitta paper filters today. They are the flat paper coffee filters that have to be opened out before placement into your coffee maker. In 1975 the Mr. Coffee brand was released and these are identifiable as the open filter with the flat bottom and slightly scalloped sides; you probably use them in your office coffee maker at work.
Debate rages in some quarters over whether the nice white paper filters which have been bleached are a health risk or impart a flavor to your coffee. Dioxin, which is a known carcinogen, is a byproduct of the bleaching process. Unbleached fliters appear less white. I have no idea if there is really any risk to your health from the bleached paper, but personally I always use unbleached paper filters on the ocassions that I make coffee with my filter coffee maker.
One alternative to get around this debate if you are a regular filter coffee maker is to buy a gold mesh permanent filter for your machine. Gold being a totally inert metal imparts no flavor to your coffee and is health-risk-free.