Chrono Trigger has 15 endings in all, most of which are available only after beating the game the first time and starting a new game under the "New Game +" option. Here is a list of the endings. You can also download a text file of them here.
Chrono Trigger Endings
1) Quick Ending: Beat Lavos from the time gate at the Millennial Fair in 1000 AD, right after starting a New Game +. You will appear at the end of time, where you can talk to characters throughout the game who speak for the many Chrono Trigger developers.
2) Frog-Leene Ending: Beat Lavos from the Millennial Fair time gate in 1000 AD after returning from 600 AD. Everybody in the world is now a frog and Marle is getting married to Frog.
3) Monster Comedy Ending: Beat Lavos from time gate in the bucket once you are able to go to the End of Time. A Nu and Ribbit harass each other while the credits roll.
4) Robo/Tata ending: Beat Lavos after going to 600 AD, right after returning from 2300 AD. While Robo and his girl are happy in the future, Tata in the Middle Ages is called upon to defeat the evil Chrono, Marle, and Lucca.
5) Miscellaneous Scene ending: Beat Lavos from the Millennial Fair time gate after getting ter Hero's Medal from Tata. Various scenes of Magus, Kino and Ayla, Frog, etc., are shown as the credits roll.
6) Cast of Characters ending: Beat Lavos from the Millennial Fair time gate after returning from getting the Red Stone in 65,000,000 BC. The cast of characters from the game appear between the credits.
7) Frog vs. Magus ending: Beat Lavos from the Millennial Fair time gate after Frog opens the Magic Cave which leads to Magus's castle. As the credits roll, Frog makes his way to Magus's castle, where they hold their final duel.
8) Reptite ending: Beat Lavos from the Millennial Fair time gate after Ayla returns to your party. Everybody in the world is now a Reptite.
9) Frog and Magus II ending: Beat Lavos after defeating Azala in 65,000,000 BC. Frog finds a secret statue of Magus, and various scenes of Magus, Schala, and Janus are presented.
10) Marle and Lucca's Slideshow ending: Beat Lavos after Schala opens the sealed door in 12,000 BC, but DON'T power up Marle's pendant. Marle and Lucca hold a slideshow and comment on characters in the game (especially the men).
11) Losing ending: LOSE against Lavos from the bucket at the End of Time. Lavos basically destroys the planet.
The last 4 endings are basically the same except for subtle differences, depending on who in the game is alive or dead.
12) Moonlight Parade ending (with both Chrono and Magus dead): Beat Lavos after defeating Magus in 12,000 BC, without reviving Chrono. Frog's curse is lifted, and neither Magus nor Chrono appear in the ending.
13) Moonlight Parade ending (with Chrono dead but Magus alive): Beat Lavos after Magus joins you. All the characters wake up at the End of Time, and then return to their respective times. Marle arrives at the Millennial Fair, where she is carried off by balloons to Death Peak. Chrono appears in the distance...
14) Moonlight Parade ending (with Chrono alive but Magus dead): Beat Lavos after defeating Magus and reviving Chrono. The same as ending 15, only Magus doesn't appear.
15) Moonlight Parade ending (with both Chrono and Magus alive): Beat Lavos after teaming with Magus and reviving Chrono. Chrono is awakened and told his stay of execution is over. He is brought to the castle where he is in fact celebrated for saving the world. Chrono and Marle attend the Moonlight Parade. If Lavos was beaten by crashing into him with the Epoch, Chrono and Marle are taken away by balloons and float over the world. If Lavos was beaten another way, Chrono's mom accidentally enters a time gate and Chrono, Marle, and Lucca leave to rescue her.