Verse 8: Wherefore he saith, When he ascended up on high, he led captivity captive, and gave gifts unto men. Paul now quotes from Psalms 68:18. Remember that the context is the gifts that he just spoke of in the previous verses. As such, whatever he is quoting is to make his point about said gifts.
The context of the psalm is in the future when when Christ will reign after taking back what belongs to him. The phrase led captivity captive in these verses speak of him triumphing over his enemies by the resurrection from the dead (Ephesians 1:18-22). Paul is drawing from the Roman Triumph which was a celebration of the success of a military commander. On the day of his triumph, the successful general would wear a crown of laurel and an all-purple, gold-embroidered triumphal toga picta ("painted" toga), regalia that identified him as near-divine or near-kingly. In some accounts, his face was painted red, perhaps in imitation of Rome's highest and most powerful god, Jupiter. The general rode in a four-horse chariot through the streets of Rome in unarmed procession with his army, captives, and the spoils of his war and at Jupiter's temple on the Capitoline Hill, he offered sacrifice and the tokens of his victory to the god Jupiter. The order of the procession would be the captive leaders, their allies, and soldiers (and sometimes their families) usually walking in chains; some were destined for execution or slavery. All this was done to the accompaniment of music, clouds of incense, and the strewing of flowers. This is most likely what Paul was talking about in 2 Corinthians 2:14-16.
Others have pointed out that Paul did not quote the passage precisely. The Psalmist actually said thou hast received gifts for men while Paul said and gave gifts unto men. So one says received while the other gave. He could be saying that what Christ received, he was now giving, i.e., grace gifts (verse 7, 11). Again, the point and the context is gifts.
Now, for years, I have taught that those captives being referred to were those who were in paradise who had died before the resurrection. Not so sure about that view now. He seems to be referring to his enemies instead.
Lower Parts of the Earth
Verses 9-10: (Now that he ascended, what is it but that he also descended first into the lower parts of the earth? He that descended is the same also that ascended up far above all heavens, that he might fill all things.) It must be noted that these verses are parenthetical. That means that verse 8 runs directly into verse 11. This verse simply speaks of Christ's death, burial, and resurrection. Unlike what the apostles creed says, Jesus did not descend into Hell, but Sheol; the place of the dead (Luke 16:19-31). Hell is where men go after the judgment, the Lake of Fire, and that has not happened yet, therefore, it was empty then and it is empty now. The first to be thrown into hell will be the Antichrist and the False Prophet (Revelation 19:20) and then the devil himself (Revelation 20:10).