It depends on the DS and the particular flash card. Some of them need the console to be flashed, others just work.
Remember, what you actually buy is a DS cartridge which can either accept an SD card, or a USB connection. If its an SD card, obviously you need an SD card reader. So yes in a manner of speaking your PC does need a special slot, if it is an SD card. Find out before you buy.
The particular card you buy will come with the software you need, which will allow you to run DS Roms. You can even get emulators for older systems, to run other kinds of ROMs.
I used to be big into Pokemon. That game could inspire addiction in me like nothing before or since. I've got over it though, dont think I could play through any of those games again. Just too similar, once you have finished a few. I used to have both a DS Lite and a DSi, and Diamond and Platinum. Finished both, bred new Pokes in Diamond to use in Platinum. Had finished the Elite 4 in platinum when I left my bag on a train (while still living in the UK). I lost my DSi and about 6-7 games, including Platinum. Still have Diamond and a lot of older GBA Pokemon games around here somewhere. Heck, I finished FireRed and beat the 8th gym in Emerald (not the Elite 4 though).
Its scary to think how much time I lost to those games.