He was born in Toronto, Ontario, Canada, after his parents emigrated from Poland and attended the University of Toronto as an undergraduate. After receiving his Ph.D. from Harvard in 1941 as Saunders Mac Lane's first student, Kaplansky was professor of mathematics at the University of Chicago from 1945 to 1984. He was chair of the department from 1962 to 1967.

"Kap," as his friends and colleagues called him, made major contributions to group theory, ring theory, the theory of operator algebras and field theory. He published over 150 papers with over 20 co-authors. He was a member of the National Academy of Sciences and of the American Academy of Arts and Sciences. He was the Director of the Mathematical Sciences Research Institute from 1984 to 1992, and the President of the American Mathematical Society from 1985 to 1986.

Kaplansky also was a noted pianist known to take part in Chicago performances of Gilbert and Sullivan productions. He often composed music based on mathematical themes. One of those compositions, A Song About Pi, is a melody based on assigning notes to the first 14 decimal places of pi.

Kaplansky was the father of singer - songwriter Lucy Kaplansky, who occasionally performs A Song About Pi in her act.

He was among the first five recipients of William Lowell Putnam fellowships in 1938.