Villa say they have an option to make the United States international's move permanent at the end of the season.
Manager Gerard Houllier said: "Michael is is a very hard-working midfielder who can score goals and is versatile."
Bradley joined Monchengladbach in August 2008 and scored 10 goals in 76 appearances in the German Bundesliga.
He has 50 international caps and played against England at the 2010 World Cup in South Africa.
Houllier added: "I saw him play four times at the World Cup in South Africa. I saw him once on TV but four times at the games for the national team, so in terms of scouting you cannot say the work has not been done."
Bradley's first club in European football was Dutch side Heerenveen, who he joined from Major League Soccer outfit Metrostars in January 2006.
I like the way we are building up the squad for the second half of the season
DLO
He established himself in the centre of the club's midfield after the retirement of Paul Bosveldt and was linked with Blackburn, West Ham, Middlesbrough and Sunderland before moving to Germany in the summer of 2008.
Bradley - whose father the US coach Bob Bradley was linked to the manager's job at Villa this summer - often plays as a defensive shield for the US but can also operate in a more advanced position.
After signing for Villa, Michael Bradley said he was excited by the test that awaited him.
"I am really excited to be joining Villa," he said. "I am proud and thrilled and can't wait to get going now.
"Everyone can see that Aston Villa that is a big club, a fantastic club, with tradition and great supporters.
"Over the past few years Villa have made big strides and are trying to challenge at the very top of the Premier League. I am excited to be part of that and I am excited to be here.
"I want to play at the highest level and test myself in the biggest of games. Whether it's on Saturday against Fulham at Villa Park or the week after, I am excited to be here and ready to get going."
Source: BBC Sport
Source: BBC Sport