What is the difference between the Black Stone and the Stone of Abraham (peace be upon him)?

The Black Stone is the stone located in the corner where the circumambulation (Tawaf) begins at the Sacred House of Allah. It is Sunnah (recommended) to kiss it for those who are able. It is a stone from Paradise, which descended whiter than milk, but the sins of the children of Adam caused it to blacken.

As for the "Hijr of Ibrahim" (Abraham's stone), there is no known place by this name within the Holy Kaaba. The intended meaning might be the Maqam Ibrahim (Station of Abraham), which is a stone upon which Abraham and Ishmael (peace be upon them) stood when building the House, as its foundations rose. Its location is now well-known, and Muslims pray two rak'ahs behind it after the Tawaf, in compliance with Allah's saying: "And take, [O believers], from the standing place of Abraham a place of prayer."