Who was Melchizedek? Was he Jesus Christ who lived during the time of Abraham? Or are they different persons?

Melchizedek and Jesus are not the same person. Melchizedek is mentioned in several other places in the Bible (most famously in Psalm 110:4 and the Letter to the Hebrews) in connection with an eternal lineage of priests. The author of Hebrews connects this to Christ by saying that the “line of Melchizedek” is a symbol for the eternal, neverending priesthood of Jesus Christ.

Why this connection between Christ and Melchizedek? To understand this, we have to look a bit more closely at the figure of Melchizedek. He shows up very suddenly in Genesis 14 after Abraham (at the time still called Abram) wins a major victory. Melchizedek offers sacrifice to “God Most High” and blesses Abraham (14:19). He then departs and never shows up again. This man of mysterious origins leaves just as mysteriously as he arrived. Because of this, he was later used by Christians as a kind of symbol. The mysterious man, this priest-king whose name means “righteous king” and who rules over Salem (which means “peace”), hints to the future High Priest and King of Peace, Jesus Christ. Melchizedek is not himself Jesus, but his mysterious nature and righteous character are something like a “hint” from the Holy Spirit to what lies ahead in the New Testament.
The “line of Melchizedek” draws upon this symbolism. Melchizedek’s origins and destination are both unknown; because of this, there is something almost “timeless” about him and his priesthood. We are not told where it comes from, or when it ends. So when Hebrews quotes Psalm 110:4, saying “you are a priest forever in the line of Melchizedek,” it is saying that the priesthood of Jesus Christ is also mysterious and timeless. It’s not like the priesthood of Aaron and the Levites in the Old Testament, which originated with Aaron and ended with the Temple’s destruction. This is an eternal priesthood.