Secret Stages:
To get Brinstar Depths, play 50 of the versus-type matches.
To get the city Fourside, play 100 versus matches.
To snag the Big Blue F-Zero Track, play 150 versus matches.
To get Kanto (you know, that Pokémon place), play 200 versus matches.
To attain Mushroom Kingdom 2, get either the Birdo trophy.
To get Superflat World, unlock Mr. Game and Watch and defeat Classic mode with him.
To get the Metal Mario Stage, defeat the All-Star mode with anybody.
To get the Master Hand Arena, clear all 51 of the Event matches.
To get Pupupuland (Kirby), complete Target Test with everybody.
To get Yoshi's Island, hit over 1300 feet in the Home-Run Contest.
To get the Kongo Jungle, finish the 15-Minute Melee with anybody.
Hidden Characters:
Jigglypuff -- Clear Adventure or Classic mode with any character.
Luigi -- Start Adventure mode with any character and beat the first level with the number 2 in the timer's seconds place. It should look like xx:x2:xx. You will then see Luigi appear over Princess Peach's Castle in Adventure mode during a cut-scene. Complete the Adventure and he will appear after the credits.
Young Link (Clone of Link) -- Clear Classic mode with 10 characters or 500 Vs. matches.
Dr. Mario (Clone of Mario) -- Clear Classic mode with Mario or conquer Adventure mode without dying.
Ganondorf (Clone of Captain Falcon) -- Clear Event Match 29.
Pichu -- Clear 200 Vs. Matches, clear Event Match 37, or conquer Classic mode with Mewtwo.
Marth -- Clear 400 Vs. Matches or use every default character at least once in Vs. mode.
Roy (Clone of Marth) -- Clear Adventure or classic mode without continuing after unlocking Marth.
Falco (Clone of Fox) -- Clear 300 Vs. Matches or beat 100 Man Melee with anyone.
Mewtwo -- Rack up 20:00 in Combined Vs. Play Time or play 700 Vs. matches. The Combined Vs. Play Time is how long you play multiplied by how many human players were in the game (CPUs don't count). So with four people playing, you only have to play five hours to get Mewtwo. If your friends aren't Melee fans, you can just plug in four controllers, turn on a stock game with four human players, and leave it running for a few hours until you've built up enough time.
Mr. Game & Watch -- Clear Classic Mode, Break the Targets, or Adventure Mode with all 24 characters.
Hidden Modes:
Event Matches 31-39 -- You'll need to unlock Jigglypuff, Luigi, Dr. Mario, Falcon, and Young link on top of clearing the first 30 Event Matches.
Event Matches 40-50 -- Unlock all hidden characters and stages.
Event Match 51 -- Clear all events.
Sound Test -- Successfully complete Event Match 51.
All-Star mode --
Unlock all 25 secret characters.