The deal to bring the 34-year-old Bosnia and Herzegovina international to Goodison Park is undisclosed but Everton have the option to extend it by a further year.
Begovic is Everton's second summer signing after Andros Townsend was brought in from Crystal Palace on a free transfer.
The former Chelsea and AC Milan goalkeeper will provide competition for Jordan Pickford.
"When I first heard of the possibility of joining Everton, I was really excited," Begovic told evertontv.
"Opportunities like this don't come around every day so I'm honoured and pleased to be here.
"The ambition is to get this club to where it belongs. We want to be challenging for those European spots, challenging for trophies and being in that upper echelon of the league."
"Jordan (Pickford) is a world-class goalkeeper," Begovic added. "I think the goalkeeping group here is fantastic.
"We're going to push each other every single day and set those standards in training.
"I'm definitely here to compete (for a place) and to try to help this team in any way I can."
Begovic is one of six goalkeepers to have scored a Premier League goal; the others being Peter Schmeichel, Brad Friedel, Paul Robinson, Tim Howard and Alisson Becker.
