If you're a Pokemon addict you probably already know about this trick where you can catch Mew in Pokemon Red, Blue or Yellow. If not, you may be surprised to know that Pokemon #151 is actually not that difficult to catch, you just need to confuse the game a little. The video below is a complete demonstration of how this feat can be done but, just to save you some time, I'll tell you what you'll need beforehand.
You'll need to be in Cerulean City. You need an Abra, which can be caught in the grass north and south of Cerulean City and you'll need a Pokemon that can put Mew to sleep; which can be your choice of Bellsprout, Butterfree or Jigglypuff at that point in the game. This trick only works if you didn't defeat certain random Pokemon trainers. So if you're using a completed game file, this probably won't work. I suggest starting a new game and get to Cerulean City. Then, watch this video for step-by-step instructions. (If you have all of the above requirements, then skip to 1:39 in the video).
NOTE: If you'd rather read a walkthrough, check out this detailed Gamefaqs walkthrough that I used when I caught Mew.
Why is getting Mew so great? Well, you need him in order to get a complete Pokedex, for one thing. Normally, the only way to get him was to win certain Pokemon tournaments that were held in Japan for the most part. Mew can also learn ANY move in the game. With various TMs, Mew can be an unstoppable force.
I've done this trick on my cartridge and it doesn't negatively affect your game at all. My only suggestion for the video is that you should beat all of the trainers in the area north of Cerulean City beforehand except, of course, for the two you need in order to do the trick. This way, you won't accidentally face the wrong trainer.
Now go get your old Gameboy and catch em' all! For real this time!