As the hands of time created The Multiverse, Error Gaster was made as the force of primordial destruction, and Ink Gaster was made as The Primordial Creation. Error Gaster was fine with his role, despite being forced to become a recluse. Eventually, he wounded Ink Gaster to almost lethal extents, before he decided to give up, and simply left. Thousands of Years Later, Error Looked for him to teach him. Error Gaster of course let Error find him, after traversing multiple traps to see if Error was worth his time. When Error got through the traps, he confronted him, then allowed him to live in his Fortress. He then "adopted" Error, not like Error cared very much. Error Gaster trained him to become more merciless and effective at combat, however Error was rather stubborn, much to Errror Gaster's annoyance. Eventually, Error Gaster began sending him on missions for him. Allowing Error to gain more skills, and find artifacts for Error Gaster.
Gaster is very apathetic. He has no sense of humor, or even happiness, really. He shows small signs of happiness when interacting with his adopted son at times, but not very much. He also enjoys playing The Godly Puppetmaster, he likes it when everything is in his control. It allows him to become rather full of himself. Unlike Error Sans, Gaster is much less prone to childish impulses. It is implied he was similar to Error in his past, but grew out of the personality. It is possible he still seeks to destroy The Multiverse, as it is revealed by him that Error Sans is still only a Demigod of Destruction. However, he might have alternative motives. He barely never interacts with anyone other than Error Sans, and a few trusted servants.
They're mostly unknown, however it's known that he does have strings and blasters. It is very likely he has many more powers than this.
His "brother." It is implied that he has some sort of Jealously towards him due to their positions, and the fact that everyone seemingly likes Ink Gaster, as The Primordial God of Creation. He generally seems to dislike him as well
He has no real care about Ink Sans, simply seeing him as something for Error to overcome, which in his eyes, will also prove he is a superior mentor and father to Ink Gaster. He however still sees Ink Sans as a pest, and isn't threatened by him at all.
He is his "son," which he cares about to a decent extent. It is possible he only cares about him to prove he is a superior father to Ink Gaster, but it's also possible that he actually cares about Error as a son, despite Error's general personality and mannerisms.
Later on, it is revealed that he has Error Papyrus in some sort of tank where he's keeping him. It is unknown if this is The Original Error Papyrus from The AskError Comic, or if it is a creation of Error Gaster. His relationship with it is unknown.