John 14:2 says in the King James Version, "In my Father’s house are many mansions: if it were not so, I would have told you. I go to prepare a place for you." In the NASB it says, "In My Father’s house are many dwelling places; if it were not so, I would have told you; for I go to prepare a place for you." The word for "mansion" is μονή, monay. It means a dwelling, an abode.
Jesus is not telling us that heaven has compartments or that we will have little places in which to live. In the ancient culture, a father's house was where the extended family lived. Rooms were often added on as the family grew through birth and marriage. What Jesus was doing was using the present day illustration of a loving, tight, family community. So, Christ is saying that He is preparing a place for us in heaven where we will dwell with God in close communion with Him, and that there is room in heaven for all whom God calls to salvation.
Please note that some commentators have alluded to the "mansions" as our future resurrected bodies that we will possess at the resurrection and accompanying our entrance into heaven. This seems possible, but the context of the verses does not necessitate that opinion.