I am 14 and I have noticed that my teeth are jagged at the end. Should I be worried or will they file down themselves? (photo)

I would recommend having your dentist perform dental bonding which would add the missing portion of your tooth back and match the tooth next to it.  This is typically very easy and painless.  This bonding will need to be replaced ever once in a while due to staining that can occur.  While you can have your dentist smooth the tooth down, it is so easy to have the bonding done I would go ahead with that route.

No need to worry. I would ask your dentist to get rid of the chipped corner either by placing a filling or smoothing the edge. Both are very minor fixes.