In Skyrim, their is a class of destruction magic that revolves around placing "runes" on the ground that trigger a certain elemental explosion when stepped on. These runes seem to just be circles with Daedric text inside them. This would make it seem that they are Daedric in nature, and I just want to know how they work. Are they similar to bound weapons and the rune is just a Draedroth that has been summoned into the shape of a rune etched into the ground? Are the runes even acknowledged anywhere in the game? What, exactly, does the rune's writing actually say?

The Daedric alphabet does not always signify the involvement of the Daedra. In Morrowind, for example, Daedric letters were routinely used for things like street signs, decoration and magical scrolls (including spells like Divine Intervention). It appears to be a popular alphabet among mages and academics, comparable to the use and prestige of Latin in real-life academic circles, but with little inherent power.

EDIT: TIL has a good introduction to the Daedric alphabet http://www.imperial-library.info/content/daedric-alphabet.

I see. So I suppose that the runes aren't necessarily Daedric in nature, most likely just the player-character trying to look smart. Which explains why such a thing would be categorized as destruction magic instead of conjuration in the first place. Thank you. Oh, and after taking a quick look it seems like the Daedric written on the runes is just a random handful of letters, which supports this theory, I guess.

I think that either there is inherent magic in some symbols and language or they can be "charged" with spells. I assume that's how scrolls work and I think that robe Elysana enchanted had Daedric text embroidered onto it.

There must be a reason for why the Daedric language is used in magic and it may just be that Chimer and some humans at least learnt a lot of magic from the the daedra. Maybe a lot of destruction as well as conjuration which was taught in both the Mages Guild and the College of Winterhold is derived from witchcraft, even if one doesn't have to serve the daedra in order to use it. I'd be surprised if Altmer used Daedric that much, but then their magic probably dates back to when they had similar ancestor worship beliefs to the Psijic Order when the daedra weren't defined as "not our ancestors". Dawn magic might be a different story.

Don't read too much into the exact text on the runes. Most of the scrolls in Morrowind said "Woe upon you", including healing scrolls and others were http://www.uesp.net/wiki/Morrowind:Scrolls. It doesn't mean that the text isn't supposed to be significant.

In fact why do the daedra have a common alphabet/language?

I think Morrowinds scrolls were originally supposed to hold a unique spell that would be spoken to release the magic. But halfway through it the devs ran out of ideas and/ or time and copy-pasted the "woe upon you" on the rest.

The symbols on the runes really don't seem to hold a meaning, or at least no obvious one. the letters are (clockwise from the top, couldn't make out the fourth) M F Q ? H T S C P. Only the one in the middle is an F for fire and L for lightning (and I guess I for ice).