The NPP includes more than just a new set of interpretations for Paul's works, it also re-examines several other historical issues and purports to throw a different light on them. In reading through material from both sides, I find that much breath seems to be wasted on detailed analysis of issues that aren't actually new or different in any way, but that a few key issues (particularly the definition of Justification) are substantially different. While some NPP adherents claim that their new view of Justification is compatible with the old, most who hold the old view find the new to be incompatible.
Full disclosure: I believe that, while the NPP has some valuable perspectives to offer Christianity, some of its premises are invalid and some of its conclusions are heretical. Still, I will try to answer here carefully. If my treatment of the facts of the case are not fair, feel free to comment.
Was 1st Century Judaism so off track?
The basis for the "new" understanding of Paul's life and teachings is really rooted in a different understanding of the religious culture of his day.
Historically, most of Protestantism has understood 1st century Judaism to be a works-based religious system under which people crossed their t's and dotted their i's in accordance with a vast set of laws and regulations in order to earn or deserve God's favor. NPP submits that Judaism in Paul's time was actually more about grace than Reformation era Protestants give it credit for: that it taught grace as the basis of salvation and works as a basis of judgement, not works as the basis of salvation.
What goes around comes around.
If 1st Century Judaism wasn't a religion of works, then Paul's discourses on it must not have been raising that as an objection. If he was actually in agreement with Judaism on the issue of works vs. grace, then the contrasts we find in his teachings between works and grace must not have been a matter of right vs. wrong religion -- as most Reformed teachings claim -- but one of two parts of the same right religious understanding being contrasted. Hence, the NPP would have works and faith working together in the effecting of salvation.
With works and faith working together rather than one being the result of the other, Wright and others from the NPP movement must also recast the doctrine of imputed righteousness. Rather than Christ's work being the sole item to the credit of our accounts, our own works will be part of our judgement. Wright does not believe that God's righteousness can actually be transferred to anyone, only that he can declare (law court terminology) people to be in good standing; their own righteousness will still be based on their own works.