All you need is a Nintendo 3DS and Pokemon Yellow on it.
That's it. No cheat devices. NONE.
If you want to be really fancy, you CAN use a GameBoy, Super GameBoy, GameBoy Pocket, GameBoy Color or GameBoy Advance with the GameBoy Color edition of Pokemon Yellow, or any other GameBoy player with the GBC port of Pokemon Yellow, but for those of you on a budget, the 3DS version will suffice.
To do the glitch, you need to have JUST ENTERED Cerulean City for the first time. If you have already battled everybody from here to Bill's house and beyond, I'm sorry, but you failed. You'll have to start over on a new game file.
You'll need to trigger certain story events as well. This glitch is a little harder than it is in Red and Blue, so please bear with me.
SAVE YOUR GAME
Then, head towards Nugget Bridge and defeat your rival.
I caught mine by having Pikachu use Thunder Wave to paralyze him, and then tossing a PokeBall. Some have had luck by using Clefairy or Jigglypuff's Sing. Again, be patient.
Head back to the Pokemon Center, heal your team and put Abra on your team. Buy some PokeBalls and then go back to Nugget Bridge. It's time to activate the glitch.
BEFORE YOU DO THAT you might want to switch to Box 2 on Bill's PC in the PokeCenter, just in case Box 1 is filled up. If your main box is full, you cannot catch Mew unless you have an open spot on your team.
Even though he is paralyzed, he will still hit you back 90% of the time.
If you are on the 3DS, hold down on your circle pad once he is inside the PokeBall but just before the struggle. I don't know why, but for once, this old trick actually kept him inside a ball for capture.
So now you should have two Mews you can train in any way you want. Mew can learn all TM and HM moves, so feel free to mix it up. You can raise a Mew that knows Bubblebeam and Thunderpunch for example, even though they are a psychic types! Raise them well, so that when it's time to send them to Pokemon Bank for Sun and Moon, they'll be ready for anything.