Gaspar, a wise and observant sage, is the Guru of Time in Zeal and the guide that helps Crono and his friends throughout their journey. After serving in Zeal with a red uniform, Gaspar was thrown to the End of Time in the Ocean Palace Incident. Prior to the expulsion, he had worked in his labs at Zeal on what would become the Chrono Trigger. At the End of Time, he somehow constructed a platform in the style of our own Victorian Era and created a separate station for the Gates, which he observed were created by The Entity for the oncoming quest to defeat Lavos. While awaiting the adventurers, he somehow came into contact with Spekkio (possibly a Nu assistant, akin to Belthasar's), giving the God of War his own room. Gaspar passed his time by sleeping beneath the center lamp post with a humorous, prominent snot bubble always present.
Throughout the adventure, he told the party (among other things) where to find the Epoch, the Clone, and what to do with a few Key Items. He also presented the finished Chrono Trigger to the party after Crono perished, at which point he was revealed to the group to be the old Guru of Time. While Gaspar is painted as a reminiscent old codger with a tired spirit, he does exhibit wry humor at times and shows caring emotion when he mourns Crono's death and delivers the song Crono & Marle ~ Far Off Promise. After the successful journey to Death Peak, he elucidated several sidequests that would empower the party, then urged the party to finish the conflict and defeat Lavos forever. The Gates ceased to exist after the mission was complete, and Gaspar, Spekkio, and the End of Time's fates are unknown.
Gaspar has a line that reads,
OLD MAN: One of you is close to someone who needs help... Find this person...fast.
Fans speculated on what this meant for years, and Nintendo once fabricated a false answer to the puzzle on its website (holding Lara's leg sidequest to be the referenced topic). The original line was simply meant to convey that talking to the party members at the End of Time would reveal specifically what needed to be done in their home era. More differences can be found at Translation Differences.
Gaspar was at one time intended to be the eighth playable character of Chrono Trigger. This plan was likely cut before the Chrono Trigger Prerelease was issued, as no extra sprites for him appear in the ROM. Toriyama's sketch demonstrates that Gaspar would have worn his Zeal robes and carried his staff in the party. The sketch, and his comment of proof, appear in the Chrono Trigger V-Jump Player's Guide.
Belthasar, Melchior, and Gaspar are three wise men who brought gifts of gold, frankincense and myrrh to the infant Jesus according to legend.
The following is from the afterword of the book "Lamb" by the author Christopher Moore. The novel itself is a work of fiction, but it is based on some truth. Moore did extensive research of both the Gospel and the Bible as a whole before writing. Here's what he says:
"Another Gospel misassumption is that the three wise men were kings, or, in fact, that there were even three of them. We make that assumption because there are three gifts given to the Christ child. Their names are never mentioned. The names Balthasar, Gaspar, and Melchior come to us from Christian tradition written hundreds of years after the time of Christ." (pages 442-3)
When the Catholic church committed those wise men to sainthood, they did so with the names Melchior, Gaspar, and Balthasar. There's no actual record in the primary sources, but all three names were canonized together at a much later date.
A couple of centuries after the Gospels were written (say, 7th or 8th century), though, the names Gaspar, Melchior, and Balthazar were traditionally used to describe the three wise men. Perhaps people were trying to "boost" Jesus' image by showing that royalty from the three major world powers were visiting him -- Balthazar would obviously have been a Babylonian name at the time, while Gaspar/Caspar would have been Persian and Melchior would have been Arabian.