The Senegal striker has signed an eight-year contract at Stamford Bridge to become Mauricio Pochettino's second summer signing.
Jackson scored 12 goals in 24 La Liga games for Villarreal last season, including nine goals in the final eight games of the season. He was named La Liga's player of the month for May.
The 22-year-old joins France attacker Christopher Nkunku as summer arrivals at Chelsea.
Chelsea co-sporting directors Laurence Stewart and Paul Winstanley said: "We are excited to welcome Nicolas to Chelsea. He is a young player with big potential, as he showcased for Villarreal last season.
"We believe he is ready for this next step in his career and look forward to him working with our new head coach, Mauricio Pochettino, and his Chelsea team-mates."
Bournemouth agreed a deal to sign Jackson in the January transfer window but he failed his medical due to a hamstring problem.
Jackson has won three caps for the Senegal national team and was part of the Teranga Lions' squad at the 2022 FIFA World Cup in Qatar.
