Hello!
Exception most commonly means a compromise. For example, if a person is selling lemonade to her friend, she may say "It is fifty cents, but I will make an exception for you." This means that she is compromising, or making an agreement, with her friend to make the lemonade a cheaper price.
In other words, an exception is when a rule is changed for a particular person/reason, but the rule remains the same for everybody else.
I hope this helps answer your question! Have a fantastic day!