Anelka returns to the Premier League following an unhappy year in Chinese football and a short spell in Italy.
Anelka left Chelsea for Shanghai Shenhua in January 2012 but joined Juventus in the January transfer window.
A move to The Hawthorns reunites him with Steve Clarke, the Albion boss who was a coach at Chelsea when the Frenchman arrived at Stamford Bridge at the start of 2008.
Clarke, who has seen last season's top scorer Romelu Lukaku return to Chelsea, said: "We are missing a bit of firepower from last year - we're trying to put that right - and to kick off with the signing of Nicolas at the start of the summer is a great boost.
"The more quality players you can bring to the club the better - and no-one can question Nicolas' quality.
"He's a player I have worked with before as I was at Chelsea when they signed him. He's got a great work ethic, looks after himself and is a consummate professional.
"If you add that to the obvious talent he's got I think he can prove to be a really top signing for the club.
"His experience with some of the top clubs in England and Europe will stand the group in good stead.
"Hopefully he will be able to inspire the other players to even greater heights than we managed to achieve last year."
Anelka, 34, has played for 10 clubs in his career and Albion are his sixth in England.