Pelias was power-hungry and he wished to gain dominion over all of Thessaly. To this end, he banished Neleus and Pheres and locked Aeson in the dungeons in Iolcus. While in there, Aeson married and had several children with Alcimede, most famously, Jason. Aeson sent Jason to Chiron to be educated while Pelias, paranoid that he would be overthrown, was warned by an oracle to beware a man wearing one sandal.
Many years later, Pelias was holding the Olympics in honor of Poseidon when Jason, rushing to Iolcus, lost one of his sandals in a river while helping someone cross. When Jason entered Iolcus, he was announced as a man wearing one sandal. Paranoid, Pelias asked him what he (Jason) would do if confronted with the man who would be his downfall. Jason responded that he would send that man after the Golden Fleece. Pelias took that advice and sent Jason to retrieve the Golden Fleece.
Alternatively, he survived until Jason and his new wife, Medea, came back to Iolcus. She slit Aeson's throat, then put his corpse in a pot and Aeson came to life as a young man. She then told Pelias' daughters she would do the same for their father. They slit his throat and Medea refused to raise him, so Pelias stayed dead.