This question comes up a lot, and so in response to this forum thread I'll illustrate the process of die-cutting on a platen press below! There are many methods for successful die-cutting; this is just how I do it. The press in the pics is a 10x15 C+P, but you can follow the same process on both larger and smaller presses. Most important is to protect the press itself, specifically the press bed and the platen.
There are also many places that will make letterpress dies (you'd order a type-high, steel-rule die, most likely from an outline created in a vector-based design program). If you have a die-maker you like, feel free to share that info on the Supplies and Sources Forum page.
Step 1: Remove the rollers
Step 2: Lock up your die, just as you would a base or form
Step 3: Put a piece of pressboard, or other hard, thin, piece of something, behind the die in its chase and on top of the press bed:
This will prevent the rule in the die from scarring the press bed; it gets pushed in a little as you cut. Here's another view:
Step 4: Remove all packing and gauge pins! I also switch to a beat-up draw sheet and save the clean sheets I use for printing.
Step 5: Protect the platen from scarring as well. I use a piece of spring steel (Item #9014K61 from McMaster-Carr), cut down to roughly 6x8, instead of a die-cutting jacket. Having a hard surface to cut against also ensures that you'll get a nice, sharp cut and a clean edge on the final piece. The steel piece gets taped to the tympan:
Note: there are various ways to determine where your die is going to hit: you can measure; you can tape a piece of transparency film to the draw sheet and slide the plate under the mark it makes; you can cut right on to the draw sheet and position the metal accordingly.
Step 6: Once you are reasonably knowledgeable about where your die will hit the platen, you can position your gauge pins. Keep them away from the plate and cutting rule so that they don't get smashed:
Step 7: Position the paper. This happens much as it would if you were printing, in terms of straightening, centering, etc.
Step 8: Once the paper is positioned where it needs to be, you're ready to cut! You can add packing behind the draw sheet so that the cut is clean through the sheet, but the shape shouldn't fall out as you're taking the sheet off the platen. I usually use one relatively thin sheet of packing, and then push the shape out on to the delivery board with my left hand as I'm feeding the new sheet in with my right.
Step 9: Enjoy the lack of wash up!