Water is slowly filtered by the ground coffee absorbing the oils, caffeine, acids and other components.

The paper filter stops the coffee grounds from falling through assuring the drink is free from grounds.
This filter can then be thrown away making for ease of use.

The paper filter was invented by Melitta Bentz of Germany in 1908 as she found the coffee of the time was over brewed and often had coffee grounds in the drink.

This simple filter has not changed much since this time and is now used the world over.

In some countries metal filters are used as these do not need to be replaced each time, but of course require regular cleaning.

A commercial filter coffee machine

Filter coffee machines are used through out the world, but have a greater prominence in North America where filter coffee is the more popular version of coffee.
Through Europe, espresso based coffee, coffee produced in a cafetiere and instant are the more dominant drink served.

The first automatic drip-brew coffee maker was produced by Bunnomatic in 1963.
Since then the design has not changed much as per the example above.