This is a significant chapter in understanding how the first century church understood the resurrection. It also contains some important early history of the events following Jesus resurrection including a "post resurrection" appearance list. The list begins in verse 5 and includes Cephas (that is Simon Peter), the 12 (that is the Apostles as a gathered group -- one assumes including the newly elected Matthias), 500 brothers and sisters at once (must have been something to be a part of), James (this would be the James known as the "Lord's half brother"), then to all the apostles -- remember the credential for an apostle (Acts 1:22) was that they were there from the beginning (the baptism of John) and were a witness to the resurrection (this is clearly a number greater than 12), and finally, Paul mentions himself as having seen the resurrected Jesus. He means his vision on the road to Damascus (Acts 9).
Verses 12 to 28 contain Paul's argument on the centrality of the resurrection to the Christian's salvation, life and future hope. "If for this life only we have hoped in Christ, we are of all people most to be pitied (19)." He compares the spread of death through the disobedience of the first humans with the spread of life through the obedience of Jesus Christ -- through Adam spread mortality but through Christ spreads life eternal.
Verse 29 is a classic hapax. A hapax is a term used in biblical studies to describe a verse or passage of the bible that only appears once and its meaning is unclear. In this case Paul is talking of "baptism on behalf of the dead." No where else in the New or Old Testament is any such practice mentioned. No where else in the scriptures is there any suggested theology for such a practice. The rule of thumb with a hapax is that they be held lightly. We understand the verses and chapters of the Bible in light of the whole (not the other way around). One religious tradition has raised this single verse to an extended and essential practice. I would suggest that we hold this, like we hold the mention of the 1000 year reign in Revelation and women keeping silent in the churches in 1 Corinthians 14, lightly and pray that one day we will understand what was intended.
Paul ends the chapter with a description of the resurrection body -- physical bodies die to be raised as spiritual bodies. Our glory in human form is fleeting but in resurrected form will be eternal. He concludes with a reminder that history is coming to its climatic conclusion -- the trumpet will sound and the dead will rise from their graves and the corruptible will become incorruptible and the perishable will become imperishable.