When you have certain rules that people have to follow, sometimes you let a few people break the rules. Letting people break the rules is called "making an exception".
For example, a teacher might let some students turn in their homework late if they have a good reason why they couldn't finish the homework by the deadline:
OK, I'll make an exception since you were sick. But next time, make sure to turn it in on time.
"No exceptions" means that you're not going to let anyone break the rules:
A: Can't you bend the rules just a little, just this once?
B: No, I'm sorry. No exceptions.