Terms were not disclosed, but the former Alabama star running back will compete with Latavius Murray to be the Raiders' starting running back.
Richardson said Tuesday that he expects to win the Raiders' starting job.
"If a guy is coming in and doesn't expect to be a starter, why is that guy here? For me, I expect to be a starter wherever I'm at. I just can't wait for the opportunity," he said.
Richardson struggled in both Clevelandand Indianapolis, but the Raiders think the 24-year-old is worth a shot.
"Physically, I think that when you get to this level, you can play it," Richardson said Tuesday when asked if he was surprised by Chris Borland's sudden retirement on Monday. "But mentally, when you sit and think about stuff or your mind is not into it, that can tear you down as a man. A lot of football players don't realize stuff until it's gone.
"I've been playing football my whole life. I've been playing football since I was 6 years old. Coach asked me this yesterday. ... He said, 'So, what would you do without football? At your age, what would you do without football?' It kind of hit me to where, 'Yeah, you're right, Coach.' That's something that you don't expect, that you don't hear from everybody."
The Raiders, who have signed a league-high 10 players in free agency, tried to sign DeMarco Murray, but he went to the Eagles. Maurice Jones-Drew retired and Darren McFadden went to Dallas in free agency, leaving Latavius Murray in Oakland's backfield. The Raiders also had signed Roy Helu to be a backup.
Latavius Murray showed flashes last year, but he is inexperienced, and new coach Jack Del Rio said earlier in the offseason that he wants Murray to work on his football IQ.
Richardson filed a grievance against the Colts last week for voiding the $3.184 million he was to make during the 2015 season before his release. He was let go by the Colts after the team signed free-agent running back Frank Gore.
The Colts acquired Richardson, the No. 3 overall pick in the 2012 draft, from the Cleveland Browns for a first-round pick in September 2013 with the vision of teaming him with quarterback Andrew Luck, the 2012 No. 1 overall pick.
The trade ended up being a failure for general manager Ryan Grigson. Richardson rushed for 977 yards on 316 attempts in 29 games with the Colts. He failed to crack 100 yards rushing in any of those 29 games.
Raiders quarterback Derek Carr tweeted a welcome to Richardson after the signing was announced.
"I was expecting this," Richardson told ESPN.com last week about his release. "It just didn't work out. I'm not saying anything bad about the program, but that program just didn't fit me at all. Me and the GM didn't see eye to eye. It was a bad marriage for us."
He entered the 2014 season as the starter, only to lose his job to Daniel Herron late in the season. Richardson fell behind Herron and Zurlon Tipton, who spent the first part of the season on the practice squad, in the playoffs. After being inactive for the divisional playoff game against Denver, Richardson said that would never happen again.
Richardson was right; he wasn't inactive for the AFC Championship Game against New England. He was suspended for two games by the Colts after he failed to notify them that he would miss a walk-through session the day before the game against the Patriots. Language in Richardson's contract allowed the Colts to avoid paying the final year of Richardson's rookie contract because he was suspended.
Richardson missed the walk-through because his wife was in the hospital with serious pregnancy complications.
Richardson, who will turn 25 in July, averaged 3.3 yards per carry for the Browns and Colts, both of whom invested first-round picks for Richardson. He said he wants his career flux to end.
"Hopefully this is my last stop, and I'm going to do whatever I can to make this my last stop," Richardson said. "Until I'm ready to walk away from the game, I don't want to leave here."
Mike Wells of ESPN.com contributed to this report.
Raiders Take Chance On Richardson
Adam Schefter reacts to the news that the Raiders have signed RB Trent Richardson.