The cinematography in The Godfather is worth discussing on its own terms, separate from everything else the film achieves.
Gordon Willis earned the nickname 'The Prince of Darkness' for his work on this film. He lit the interior scenes so that the men's eyes are often in shadow — a technique that conventional cinematography would have rejected as technically defective. Actors' eyes are supposed to be lit. You're supposed to be able to read their faces.
Willis understood that the Corleones needed to be partially illegible. The moral obscurity of the characters is mirrored in the formal obscurity of their faces. When Brando's Vito is in full light, it means something specific. When he's in shadow — which is most of the time — it means something else.
Paramount initially rejected the footage. They thought Willis had made technical errors. Coppola held the footage. The darkness stayed. It became the most imitated look in American cinema history.