The centre back joins Celtic from RB Leipzig, and has made 195 appearances in the Bundesliga.
The 32-year-old has also had spells with Borussia Monchengladbach, 1899 Hoffenheim and Italian Serie A side Fiorentina.
Compper admitted the appeal of winning silverware influenced his decision, saying: “[Celtic] are a club who play for titles and that is something which has been lacking throughout my career.
“I was playing in good teams but never won a title. This is something I’m really looking forward to, having the possibility to play in big games for big wins. Celtic is a club that is always going for the big wins.
“I’ve played in important games in Italy and Germany. I was in a cup final in Italy and we lost, unfortunately, but I hope I can add silverware to my career here.
“Celtic are obviously one of the biggest clubs in the world. The tradition, history and the incredible supporters are what have stood out to me.
“They are a very big name, even in Germany. Everyone in Germany knows about the ambience in the stadium and it’s seen as a football-crazy place. Celtic is a big name not only in Germany but worldwide.”
Compper will be paraded at half-time during the match between Celtic and Partick Thistle this evening.