Paul only explicitly discusses Adam for about 10 verses in Romans 5 and a verse or two in 1 Corinthians 15. So the significance Christians find in these verses is disproportionately large to the attention they actually get in Paul's own writings and the Bible as a whole. The reason is that in these few verses Paul points to Adam as the reason why Sin and death are in the world.
As far as death, 1 Corinthians 15:22 puts Paul's thinking succinctly: "as in Adam all die, so in Christ all will be made alive." Because of Adam, all human beings die. This astoundingly brief comment, and the more detailed explanation in Romans 5, have had immense influence on key Christian thinkers like Augustine and John Calvin--and through them on the majority of Christians today.
The way Adam introduced death into the world, Paul says, is through the fact that he introduced Sin into the world: "sin entered the world through one man, and death through sin" (Rom. 5:12). Paul's sense of how this happened is vague to us. In fact, it may very well have been vague to the Romans themselves (cf. 2 Pet. 3:16). But perhaps the best explanation is that Paul saw Sin as a power that came over the creation as a result of Adam's disobedient act in the Garden of Eden.
First, it is clear that Paul saw Adam's sinful act as the cause of the rest of humanity becoming sinners, since he says in Romans 5:19, "through the disobedience of the one man the many were made sinners." In some way that Paul never really explains, Adam introduced into the world something he calls the "law of sin and death" (8:2). The result is that I now find I have a "law at work in the members of my body" (7:23). The result has been that "death came to all men, because all sinned" (5:12).
But Paul indicates that this power has also somehow enslaved the rest of the physical realm as well, in addition to my "flesh," my physical body. Again, Paul is not clear at all on how it all works, but it would seem that God subjected the rest of the material world to corruption and decay at the same time that humanity came under the power of Sin (cf. Rom. 8:20). Similarly, Paul believed that the rest of the creation would be freed from decay at the same time that our physical bodies were transformed whether by resurrection or when Christ returns from heaven...