[edlinfan]

The simplest way to deploy your homebrew ROMs to a real NES is to use something called a "flash cartridge"[1]. This is equivalent to a real NES cartridge except it is backed by rewritable flash memory instead of a permanent ROM. It contains all of the necessary glue logic[2] so the console can't tell the difference. You can rewrite the flash if you connect the cart to your computer via USB.

The magical thing to google if you want to do your own research is "NES flash cart".

The "EverDrive" that aji suggests is, indeed, an example of a flash cart. It may or may not be the best one for the NES - I honestly don't know, all of my experience is with the GameBoy.

dpflan's approach of erasing and reprogramming a REAL NES cart is also possible, if you enjoy pain and suffering in the name of being lavishly historically accurate. (I know I do sometimes.)

  [1] https://en.wikipedia.org/wiki/Flash_cartridge
  [2] https://en.wikipedia.org/wiki/Memory_management_controller