Allardyce has been out of work since leaving Crystal Palace in May.
“The ambition of the club, allied to all the fantastic history, has proved to be key for me,” he said on being announced as the club’s coach.
“Obviously, the club has gone through a difficult spell and hopefully I can put that behind us as quickly as possible and start looking upwards again. I think that whatever’s happened before we’ve just got to all rally around together. Our will to do it all together would be my way forward. I feel really enthused and energised to come in as manager,” he added.
Former coach, Koeman, had guided Everton to seventh place last season but was sacked after a 5-2 home defeat by Arsenal on 22 October.
Despite spending more than £130m on new players in the summer and seeing the return of former boy wonder, Wayne Rooney, they have struggled without last season’s top scorer Romelu Lukaku, who was sold to Manchester United for £75m in July.
Under the U-23 coach David Unsworth they had faced difficult times too and had won two of their last eight matches.
His final game before returning to his former role was Wednesday’s 4-0 Premier League victory over West Ham at Goodison Park, which Allardyce watched from the stands.
Sam Allardyce with Everton owner Farhad Moshiri
“Sam understands the long-term ambitions we have for this great club and I know he is a man who gives it his all and is focused 24 hours a day,” added Everton’s majority shareholder Farhad Moshiri.
“I would like to put on record our thanks as a club to David Unsworth for stepping up when we needed him over the last few weeks. He is a real Evertonian, embodying the commitment, pride and passion we all feel for the club.”
Everton are now 13th in the league and face Huddersfield on Saturday.
Sam Allardyce is a cat with nine lives.