This is a guide on how to catch Mew in Red, Blue, and Yellow. It is 100% true and has been proven many times over. Don't believe me? Follow the guide and do it yourself. You'll end up with a Mew.
Okay, first off, do NOT fight the two trainers shown here. Unless you're an idiot, you should know that I mean the trainers that your guy is looking at. You can use other people, actually — click here for more info.
Very important: To do this, you must have a Pokémon that knows Fly. Teleport works too, but Fly is more convenient.
Position yourself as shown, making sure that the old guy in the above picture doesn't see you. I would highly recommend saving.
Okay, here's the hard part, and it's actually not that hard. Take one step forward, but as you do it, press Start. You should end up like this. If the trainer saw you, turn off the gameboy, reload, and try again. Name of trainer not shown to protect the innocent
As shown, select your Flying Pokémon and have it Fly to Cerulean City.
The trainer will now see you. It's okay, because you'll fly away before he challenges you. Henceforth, the Start button will not work. That's okay, just move along.
Once in Cerulean, head north up Nugget Bridge, as seen here.
Go to the right once you get to the top, and stop where the guy in the picture is. If you talk to the kid while in the latter position, your game will freeze.
Take one step forward and be challenged by the Youngster. Defeat him, obviously.
Now, fly to Lavender Town. Head west towards the route you started on in the first place. As soon as you try to leave, the menu will pop open. Close it in any way you like (Fly, Teleport, or just hit B). You will be challenged by a Level 7 Mew.
Catch Mew and do a happy dance. =D
If you've already beaten the trainers:
You can use other people for this trick. For the bald guy, you need a trainer that is positioned so that he or she is exactly one square out of view. You just have to take the step and open the menu and fly away to initiate the trick. There's a guy in the grass right by the Nugget Bridge, for example. I think one of the Rockets in Mt. Moon also works.
As for the second person involved, you need to fight a Pokémon with a Special stat of 21. The Youngster's Slowpoke has this, which triggers the fight. If you've already beaten him, you can use the Swimmer in Misty's gym (the person in the water). If you've already beaten that person, go down to that route to the east of Vermilion City (Route 8, maybe?) and fight a Drowzee. Then go back to Lavender and trigger the battle. If you don't get Mew, just try again. There's a pretty high percentage of Drowzee with a Special stat of 21 on that route. Alternatively, you could fight a Ditto, with your Pokémon having the Special stat. Have Ditto transform, then beat it and go to Lavender.
There's a guide for this trick on GameFAQs.com — search for Pokémon Red, Blue, or Yellow and you'll find the guide. It has more info than this one does, although no pictures =)
How to do this trick BEFORE you get Fly
First off, you need an Abra, which means you need to be able to go to that route and get it (it's the one north of Vermilion). Catch one, then go back to Cerulean and heal at the Pokémon Center (even if you're fully healed). Go up Nugget Bridge and trigger the event using the dude in the grass. Then Teleport back to Cerulean, go and beat the Youngster with the Slowpoke, and go back to Cerulean to get yer Mew. =D Fwee, free legendary to pwn the rest of the game with! XD