Hello.
I have been recently playing The Elder scrolls skyrim. I have a question, when I use the magic spell Firebolt the enemy sometimes catches on fire and takes damage from it. If I use Firebolt again on an enemy already on fire, will I do extra damage with Firebolt?

Yep, it will work again. The initial strike will cause more damage plus they'll stay on fire a little longer.

From UESP: "Firebolt: A blast of fire that does 25 points of damage. Targets on fire take extra damage."

He's asking if he hits an enemy with a firebolt that's already on fire from a previous firebolt strike, will it do more damage.

This extra effect evidently also works with the flames spell - i.e. using it in bursts, rather than a continuous stream.