A nonshooting foul in basketball is a foul that is not a shooting foul. A shooting foul occurs when a ball handler who is in the act of shooting is fouled. Therefore, a nonshooting foul is any foul that is not on a shooting player.
Depending on the foul count of the team which fouls, a nonshooting foul can result in free throws. Remember that once a team fouls a certain number of times the other team begins being rewarded with free throws after fouls. If a team is in the bonus for fouls, a nonshooting foul called on the other team will result in free throws.
There are many nonshooting fouls. A few examples are traveling, blocking, and charging.
If you're looking for more information on fouls, check out these pieces: