1Co 15:29 NASB - "29 Otherwise, what will those do who are baptized for the dead? If the dead are not raised at all, why then are they baptized for them?"
What is Paul talking about here?
The big issue here is that some of the Corinthians in the church are denying the resurrection from the dead because this was totally inconsistent with the philosophy of the Greek culture which looked upon escape from the “prison” of the physical body as a good thing. Resurrection would be the last thing a Greek would want. Life after death , they desired but not resurrection.
So the church has significant doctrinal problems not to mention struggles in sexual immorality, internal bitter church fights, divisions between rich and poor, and disorderly worship services. The entire letter is filled with rebukes and warnings. Their denial of the resurrection is most important for in this they are losing the gospel itself (1 Corinthians 15:1-4)
Paul is pointing out how many things they do are based on the hope of physical resurrection.
1. The preaching of the gospel is useless if there is no resurrection (1 Co 15:14)
2. Paul and the other Apostles are all false witnesses of God (1 Co 15:15)
3. No resurrection means that Christ was not resurrected (1 Co 15:16)
4. You are still in your sins and your faith in Christ is useless (1 Co 15:17)
5. The dead have then perished (1 Co 15:18)
6 . To hope in Christ for only this life is a bad decision (1 Cor 15:19)
7. If there is no resurrection why do they baptize the dead? (1 Cor 15:29)
8. Why does Paul and his team face danger every hour? (1 Cor 15:30)
9. If the dead are not raised what profit is there for people who suffer? (15:32)
10. If there is no resurrection from the dead let us live lawless lives (15:32)
So this mentioning of the baptism for the dead is listed here of one of many reasons why denying the resurrection wipes out their Christian faith and practice. This is not just one minor belief but critical to the entire Christian worldview.
So want is this baptism for the dead and why don’t we co it today like the Mormons?
There appears to have been a baptism of the dead practiced by pagans. Later it appears the heretical Gnostic groups. Either Paul is arguing that the common cultural practice in their society is that people are baptized for the dead or they are even still being baptized for the dead. This practice is inconsistent with denying the resurrection since the practice must have been done in some way that would help them, which it cannot do if they are not raised from the dead. 
Paul does not decide to rebuke them for this practice since they are denying the resurrection, which is a more important doctrinal issue. So silence here would not be approval but simply noting that something they do accept and do is useless if the dead are not raised.
Other explanations are:
1. “baptized for the dead” is a reference to people professing at their baptism their conviction that the dead are raised.
2. “baptized for the dead” was a way of speaking of people being martyred (Mk 10:38; Lk 12:50).
3. “baptized for (above) the dead” Early worship sometimes took place in among the tombs of the dead and therefore is talking about when people were baptized over the tombs of the dead.
4. Some saw here the idea of baptism given to people just prior to death.
So because we have only this one verse as part of an argument on the topic of resurrection it is difficult to be dogmatic. But there are many orthodox answers that do not require heretical teachings.
We need to see that the resurrection is critical to our faith and that much that we do as Christians is based on our joyful anticipation of glorified resurrection into the Kingdom of God. Let the gospel be shown in how we live as well as speak to all those around us.