Smith, a lifelong Villa fan, leaves Brentford, while Terry moves into coaching having just announced his retirement as a player.
The Midlands club had been interested in Celtic boss Brendan Rodgers, who was said to be on a shortlist of preferred candidates.
However, no official approach was made and the Championship side have decided to go instead with the 47-year-old former Walsall boss.
Aston Villa chief executive Christian Purslow said on the club’s official website: “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.”
Terry, 37, finished his playing career last season with defeat in the Championship play-off final with Villa under former boss Steve Bruce, who was sacked last week.
Rui Faria, a long-time assistant coach to Jose Mourinho, had also been linked with the vacancy, as well as Thierry Henry.