The nut straight in poker is the highest possible straight.
This term is only used in games with community cards. In such games, a player must play some of the community cards, and hence it is often clear what cards they would need to hold in order make the nut straight. (In games like seven-card stud, the nut straight is always just an ace-high straight).
Holding the nut straight does not necessarily mean a player holds the nuts, but it usually implies that. Simply put, the nut straight is the nuts on any unpaired (or untripped) board that lacks three cards of any one suit.
As an example, in a game of hold 'em, if the board were:
The nut straight would be an eight-high straight, made by a player holding a 5 and 7 of any suit. A player holding a 2 and a 5 would have a straight, but not the nut straight.