The second-tier English club also appointed Jesus Garcia Pitarch as sporting director as the former European champions moved to replace Steve Bruce who was sacked last week.
Terry, 37, who spent the bulk of his career with Chelsea, captained second-tier Aston Villa last season.
Capped 78 times by England, the central defender recently rejected a chance to play for Spartak Moscow, saying a move to Russia was not right for his family.
His last game as a professional was Villa’s 1-0 Championship play-off final defeat by Fulham in May — a loss that denied Villa a return to the Premier League.
But his club career will be best remembered for his time at Chelsea where he won five Premier League titles, five FA Cups, as well as the Champions League and Europa League, although he did not play in the two European finals.
Smith joins from Brentford who are just one place off the play-off places while Vill are marooned in mid-table.
“These three appointments represent the start of a new era and direction at Aston Villa and we are delighted to secure their services after an extensive search,” said Villa chief executive Christian Purslow.
“Dean has a clear and successful coaching philosophy as well as a real understanding of Aston Villa Football Club. He is also knowledgeable about the Championship.
“John is one of the most decorated players in English football, has an affinity with the club and is ready to make the next step in his career working alongside Dean.”
Villa had been linked with a move for former French star Thierry Henry but he has been lined up for the coaching job at Monaco should Leonardo Jardim be axed.
Pitarch had also been sporting director at both Valencia and Atletico Madrid.
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