Chrono Trigger has 12 official endings, based on the in-game strategy guide. However, many of these endings have variants, and there are some that are not documented at all. This page lists some of the endings I am aware of, and how to reach them. The endings are listed roughly in the order you can reach them in the game (note, the point in the game when you defeat Lavos determines the ending).
If you lose in one of the final two battles, instead of the usual game over you get an even more depressing ending; a clip of Lavos rising in 1999 AD and destroying everything in sight. This ``ending'' ends with the quote, ``But... the future refused to change''...
There are two ways to reach this ending. First, take the right telepod in the Millenial Fair as soon as possible; if you don't talk to anyone, Marle and Crono will go together to face Lavos. This is not an easy battle because of the two-party-member handicap, however it's easier (IMO) than the second option: beat Lavos at the prophecy scene in the Undersea Palace. If you make it past the first form, then it's not hard, but Lavos here has 30,000HP (not the usual 10,000HP) and as far as I can tell, is faster than usual in its attacks. If you lose here, the story will continue with the prophecy being fulfilled; if you succeed in the first battle, and die in one of the final two battles, you get the usual ``Day of Lavos'' ending.
In the Prophecy scene, I was able to defeat Lavos (Strong) with Crono, Lucca and Marle, using a massive supply of Elixirs and Megalixers, and with all party members hasted. Follow the hint in the Boss strategy guide: devote a character to healing, i.e. Marle! Also, status protection is required (particularly against Chaos). This was the hardest battle I fought in the game.
Once you complete this ending, you enter the Developer Room, where the old man greets you with ``Good job! You finished the hardest version of the game! This is the development room for the Chrono Trigger game. Take a walk around and hear what the team has to say...''. After talking to everyone, you're treated to caricatures of several of the lead members of the team, before getting an ultra-fast version of the credits, sped up appropriately for you since you breezed through the game :)
One final note: it is possible to fight Lavos at the beginning of the game with only Crono. It's quite a bit of fun, I recommend trying it!
Beat Lavos after Yakra (Cathedral), but before returning to Guardia castle (use the right telepod). The end returns to Millenial Fair but everyone's croaking like a frog and comments on Marle's costume. When you return to castle, King has dug out old wedding album from 400 years past. Leene's marrying... WHO? Ends with Marle screaming, followed by ribbit.
Beat Lavos using the bucket as soon as you first arrive at the End of Time. A Nu and Ribbit running around in credits; the Nu is trying to get some sleep, and the Ribbit keeps tormenting. Quite amusing! ``The End'' text has a pair of Z's over it.
Beat Lavos via the telepod, after crossing the bridge in 600AD and before finding Tata/receiving the Hero Medal. In the first half, Robo goes to 2300. Scene similar to Millenial fair occurs: a bell rings, Robo bumps into pink robot, etc. In the second half, Hero Tata goes to face ``Magus'', when arriving at the end of castle he finds Crono/Marle/Lucca waiting, Crono laughs. Crono the dark mage?
Beat Lavos after getting the Masamune and hilt, but before seeing Melchior or going to prehistoric period. Random scenes occur, mostly containing people we have not yet met in the game (f.e. Ayla).
Beat Lavos after the prehistoric party and recovery of the gate key. This gives movie-style credits which are relatively dull.
Defeat Lavos immediately after giving Frog the repaired Masamune, before opening the magic cave. Frog must be a member of the party facing Lavos here, so make sure he's levelled up reasonably. In the first half: a scene of Lucca fixing Robo occurs in Lucca's house; Crono is asleep in the corner throughout the scene, and Marle says not to disturb him; Frog is also present, says he must attend to something, and trips over a pile of books heading toward the door. This scene is depicted in the OSV.
In the second half: Frog goes to challenge Magus in castle; scenes of Slash & Flea, Ozzie, occur, then Magus himself appears, stating that whoever wins the battle will rule the future. During the credits, sounds of the battle are heard. Ends with a silhouette of a caped individual atop the castle, cape blowing in the wind.
Defeat Lavos immediately after defeating Magus for the first time. Same as the beginning of the game, at Millenial Fair, except everyone's a reptite! The green jogger at Millenial Fair is actually human. Ends with reptite Crono running into reptite Marle. ``The End'' falls on Azala at very end, heh.
Defeat Lavos immediately after the fall of Tyrano Lair, via the right telepod. You get scenes that (mostly) turn to gold, with credits interspersed; Frog standing at the secret statue in the Cathedral; Frog laughing in a cafe; Magus proceeding through the Undersea Palace to face Lavos, alone. Magus states, at the end: ``If history is to change, let it change! If the world is to be destroyed, so be it! If my fate is to die, I must simply laugh!!''.
Defeat Lavos from Leene Square, immediately after Schala enters the sealed door leading to the Undersea Palace but before your pendant is powered up. Lucca and Marle host a ``slide show'' where they talk about all the men in the game. This is the only time in the game where Crono speaks up, when he interrupts the slide show at the end.
This ending does not allow you to save a system file. Defeat Lavos using the bucket immediately prior to the prophecy scene in the Undersea Palace. You get the same scenes that turn to gold, but without the credits. Cannot continue past ``The End'' to save a system file. In this ending, interestingly, Magus the prophet is revealed by Janus's cat Alfador -- who responds to the prophet (remember that Alfador only likes Janus?)
Defeat Lavos after the prophecy scene (this of course requires that you lose against Lavos (Strong) in the prophecy scene), but before reviving Crono. You end up at the End of Time, where everyone abruptly departs except for Lucca and Marle. Later, Lucca and Marle are at the Millenial Fair when suddenly a gate opens up, Gaspar comes out with the Time Egg followed by everyone else, who are trying to get the Time Egg back. At this point you find out (if you didn't already know) that Crono can be revived. Has an... odd... image of Crono and Marle at the very end.
Defeat Lavos after reviving Crono. The standard Millenial Fair - moonlight parade is played, then everyone says their good-bye's and departs through the gate at Lucca's invention. Then, normally Crono's cat(s) turn up followed by his Mom, who chases the cat(s) through the portal just before it closes.