Pokemon Fire Red is a remake of the original Pokemon Red. In Pokemon Red, the only way to obtain Mew was by attending an official Nintendo event, where it would be transferred using a link cable. In Fire Red, there is a way to catch Mew.
Ignore the gambler in front of the house between Lavender Town and Saffron City. Do not try to walk in front of him.
Enter the house. Save your game. Walk towards the gambler and press the "Start" button just as you get in front of him. He should see you, but the Start menu should override the conversation box.
Select a Pokemon with the "Fly" ability. Fly to Cerulean City.
Walk to "Nugget Bridge" and talk to the 5th trainer (he will be facing the upward direction while all of the rest will be looking down).
Battle and defeat him.
Fly back to Lavender Town. Walk towards Saffron City. A menu will pop up.
Press the "B" button to exit. This will put you in a battle with Mew. If you do not catch him now, you will have to start the process over again.
A simpler way of getting Mew In Fire Red is to have one transferred from a Toys R Us Wireless Connection Center. You can also get Mew by using the cheat device Action Replay. Mew's Action Replay code is 59f09908-11e58618-B751bdf4-95cef4cc.