The 6ft 1ins stopper becomes Mark McGhee's first signing as Dons manager after joining on a free transfer.
Nelson did not play for Norwich during his one season with the Canaries, who are managed by McGhee's former Dons team-mate Bryan Gunn.
The 27-year-old was previously a regular with Leyton Orient and Brentford.
Nelson is unlikely to feature in Thursday's Europa League qualifier against Sigma Olomouc as McGhee is trying to sign Amiens defender Reda Johnson as his one wildcard addition to his squad.
He told the club's official website: "I am delighted to have signed for a club of the size and stature of Aberdeen.
"My agent received a call from Mark McGhee at the end of last week and it's great to have sorted out a contract so quickly.
"I'm already looking forward to meeting my new team-mates and can't wait to see the club in European action this week."