Letterpress is a kind of relief printing (as opposed to offset, intaglio, etc.) using a printing press and movable type. The type itself may be individual letters, “sorts” for text composed by hand, or the text may be composed on a composition caster or line caster. Wood type is often used for large characters, especially in posters, and metal plates can simplify reproduction of whole pages.
Photopolymer plates have also found some acceptance among letterpress printers; however, at the Peasandcues Press, I use metal type and sometimes wood type, all set by hand. I’m not opposed to using line-cast or composition-cast text, but I haven’t had occasion to do so. I don’t and won’t use photopolymer plates for printing text, but I have used them for graphic art. It’s a promising medium for that sort of use, but my heart is still with traditional relief-print methods, such as woodcuts, linoleum cuts, and collagraphs.
For what it's worth, the press itself is not “a letterpress.” It may be a platen press, a cylinder press, an iron hand press, or some other kind of machine, but “letterpress” is a process, not a press. I print broadsides on a Vandercook 4 Proof Press. It’s a flatbed cylinder press. For smaller pieces, I generally turn to a treadle-powered Chandler & Price 8 X 12 platen press.
And finally, while I don't pound my type into the page, I have to admit that the only real "kiss impression" to come out of my shop was Marquita's. Lucky paper!