Christoffel attended the Jesuit Gymnasium and Friedrich-Wilhelms Gymnasium in Cologne and studied at the University of Berlin with Dirichlet, among others, where he received a doctorate in 1856 for a thesis on the motion of electricity in homogeous bodies. In 1859, Christoffel became a Privatdozent at the University of Berlin. In 1862 he was appointed to a chair at the Polytechnic School in Zurich left vacant by Dedekind. After moving to the Gewerbeakademie in Berlin (now part of the Technical University of Berlin) in 1869, Christoffel became a professor at the University of Strasbourg in 1872, where he remained until retiring in 1894.
Christoffel worked on conformal maps, potential theory, invariant theory, tensor analysis, mathematical physics, geodesy, and shock waves. The Christoffel symbol and Schwarz-Christoffel mapping are named after him.