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I am trying to understand exactly what a "found" enchanted item actually does.

For example, an enchanted item may say "Item powers: Cast when used: Fireball"

There is a "circinate" spell "Fireball" that one can purchase. It is said to cause (1-1) +(1-7) x level damage every round for 1 + 1 x level rounds.

Is this what the enchanted object does? In particular, does the effect of the enchanted object increase with one's level?

When you find an enchanted item, you can either experiment with it to see what it does (equip, use) or go to the Mages Guild and have the item identified. Usually the identifier is a lady with a floating orb, but the sprite might differ from town to town and from region to region.

I have never bothered to check but I assume that the spells of random enchanted items from dungeons are identical to the default spells. I think that using a spell from a magic item does not count towards increasing the respective magic school skill of the player, and it might similarly have only basic effect values without taking the player's level into account.

One more note though is that using items that cast spells wears them down slowly, and normally there is no way to repair/recharge them (without cheatmode). Items that grant passive bonuses like skill increases on the other hand do not wear down as quickly.

[Edit] For a bit of more useful information, the UESP Wiki here states that

• The magnitude and duration of the effect on a magical item depends on the player's level, as it does for spells.