Letterpress was the first printing method on the planet--the method that Gutenberg invented in 1450.
Currently, printers use three types of letterpress: platen-type printing, rotary-press printing, and flat-bed letterpress.
Platen-type letterpress orients the paper to be printed flat on the platen. The inked printing plate presses directly down onto it, resulting in the beautiful, textural finish of classical letterpress (business cards and invitations, for example.)
The rotary version (used very recognizably by the newspaper industry) can print both sides of the paper at once very quickly.
The flat-bed method, involving an inking roller and either a horizontal or vertical bed, is so slow that it's becoming quickly outmoded by the other two styles.
The movable type that is the historical basis for letterpress printing has been, for the most part, replaced by full plates laser-cut to a computer-generated pattern from photopolymer material.