I'm going to cut right to the chase. See the below screen shots:
Yes, your eyes aren't deceiving you! I was playing Super Mario Brothers on my phone! To do something like that, you need something that we call an emulator, and together with that, you need the actual ROMs, which are the games themselves.
The emulator I'm using here is a very nice, and most importantly, free one called John NES Lite (NES Emulator).
Get it here.
The free version works just like the paid for one, but it is ad supported.
It includes several customization options. For example, you can hold the phone portrait or landscape, you can alter the size of the buttons on screen, you can take screen shots, etc.
Now the only thing you need to actually play some games on it, would be the games themselves. I am not sure how the copyrights and that kinda stuff works for games on the NES. I take it, if you own the game, you can download it without worrying.
To get some games, simply google "nes roms", download the roms, copy them to your phone, and the app should automatically scan your phone for new games :)
Ahh, nostalgia! Enjoy!