Kelly, who began his career at hometown club Liverpool, left Crystal Palace this summer after an eight-year stay at Selhurst Park.
The 32-year-old, capped once by England in 2012, told the official West Brom website: "This is an amazing opportunity for me to show the manager, my new team-mates and the club's supporters what I can do for the team moving forwards.
"The size of the club helped convince me to come here and sign. I believe this club should be in the Premier League.
"Going forward, I want to help the club reach its true potential. I feel like, at a club the size of Albion, it's a great opportunity for me to help push the team forward with the attributes I believe I have."
Albion boss Steve Bruce said: "Martin is a defender of proven quality who has plied his trade at the highest level for the vast majority of his career.
"He has great experience of the game and will offer further competition for a place in the heart of our defence, as well as the option to play at full-back.
"He is a model professional who will no doubt be of benefit to the squad both on and off the pitch."