The 29-year-old Bosnia-Herzegovina player has signed a "long-term deal" with the Cherries 10 years after signing for them on loan when they were in League One.
Begovic made 17 league starts in two seasons at Stamford Bridge - but just two as Chelsea won the league.
"I am coming into an important part of my career," he said.
Begovic, who joined Chelsea on a four-year deal from Stoke City in July 2015, added: "I want to show people what I am capable of.
"Being part of a title-winning team at Chelsea was amazing but I felt now was the right time to move on because I want to be playing regular football."
Bournemouth secured a third successive season in the Premier League after finishing ninth in the table.
"Asmir is an outstanding goalkeeper who has proved himself capable of performing at the highest level over a number of years," said Bournemouth boss Eddie Howe.
"He is a leader as well and exactly the kind of character you want in the team."
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Bournemouth have signed keeper Asmir Begovic from Premier League champions Chelsea for an undisclosed fee.