I have a couple of weapons that have some powers, a sword that deals frost damage and a bow that deals fire damage. I found both of them. I noticed last night that each one has a "health" bar associated with it, that appears to be diminishing with use.

Are these weapons rechargeable? Or are they eventually going to become normal weapons with no enchantment?

You recharge them with Soul Gems

Ooh I see. That's what all those things are for

You will need to use a weapon enchanted with soultrap or a soultrap spell to fill a soulgem.
Only black soulgems can hold souls of any of the playable races.
The rest are white souls and can be trapped in normal soulgems.
A soul will go in its lowest available container, a petty soul will go into a petty soulgem if you have one, a higher one if you only have those.

Soulgems are used for creating enchantments and for recharging them. Both require filled soulgems, empty ones can't be used till they're filled with a soul.

Hmm yeah I'm just starting to learn all this I tried "using" a soul gem while holding the enchanted weapon but obviously no go. I guess I will have to learn that spell from someplace, and from there it will become more obvious. Do I also need to learn a spell to fill a soul gem? Are these soul gems filled with souls of creatures that I myself need to kill?

Yes.
You need a weapon with soul trap on or the soul trap spell, and empty soul gems.
You can also buy filled soul gems, but thats more expensive.

On xbox you need to select the weapon in your inventory and then use right button to recharge.