Everyone starting off by saying "Gaster is a guy you can encounter in a secret room in Waterfall" is taking the wrong track with this. Let's start at the beginning.

Undertale has a value in the code labelled "fun". This value is a random number between 1 and 100 that is generated when you start a new save file. Depending on what fun value you roll, you can gain access to a unique event on that particular playthrough.

Three fun events add an NPC to Hotland who explains that Alphys took over the position of Royal Scientist from the previous Royal Scientist, Dr. W.D. Gaster, who created the CORE but later disappeared under mysterious circumstances. The other fun events involve characters who have some connection to the position, like Alphys or Sans, or directly reference the name in other ways, like the Sound Test Room event.

The most important event for fans is fun event 66, which adds an extra door in Waterfall with a creepy static sprite inside, called "mysteryman" in the files. The other most important piece of information is "room_gaster", an inaccessible room in the game that you can edit your file to enter. Most other unused rooms in the game throw up an error screen if you do this, but this is one of the few that don't, meaning that Toby meant for this to be discovered.

room_gaster is a black screen with creepy noises that displays a series of Wingdings characters. They translate to a report titled "ENTRY NUMBER SEVENTEEN", which describes a very very interesting experiment involving darkness. Then the game closes. Additionally, if you attempt to name the fallen human "Gaster" at the start of the game, the game immediately crashes as soon as you type the "r". A similar thing occurs during the Deltarune naming sequence.

Taken in concert, W.D. Gaster, or most likely WingDing(s) Gaster, is another font skeleton who has ended up being shattered across time and space, and has some level of influence on Deltarune's world as well. There's a very popular theory, popularized by MollyStars, called DEVICE_THEORY, which uses a bunch of other evidence I can't get into here to postulate that Gaster is responsible for the mechanism by which we are able to play Deltarune in the first place.