If someone insults you, can you just brush it off or move on?

Depends on who it is, the circumstances, and the insult.

Sometimes. Sometimes not.

Depends on who has said it and the consequences. 

It depends on who insults me. For example, at school, my friend insulted me about my acne and how I would be the perfect model for acne cream. I didn't shoot him down because he either a) didn't realize that it was a sensitive subject for me, or b) was payback for how I accidentally insulted him in the morning. But if a non-friend insulted me, I would shoot him/her down with sass and class, no matter how light the insult was.

I've found it's better just to brush it off and get on with your life because all it is is words. It's only going to get to you if YOU let it. If you snap back with another insult or get physical it is only going to cause a fight more than it's worth.

Depends on what it is and who it is from.  I have made many a nemesis by laughing in their face when they insult me.  I usually don't worry; anyone who genuinely insults me is not worth my time, and most of them have either dropped out of school, or can't get a job or traineeship and are therefore stuck at school.  Not that there's anything wrong with dropping out of school to do a traineeship, but the kids who tend to drop out at my school are not very nice.

Depends upon the weight I value the person's personal Criticism.

A long time friend whom really knows me carries more weight then a stranger or even a mild acquaintance.

I am very honest about myself, so if I saw any truth to the insult I WOULD acknowledge it but that might be as far as I take it. It wouldn't control me or have an altering affect on my mood.