Q:

In Alien, what were the Xenomorph's two mouths used for?

The Xenomorphs in the Alien series had two mouths, one inside the other:

I remember the inner mouth being used to attack someone, but never any direct footage of the outer mouth being used. What was the purpose of the two mouths? Was this ever established, or was it simply left to our imaginations?

A:

Wow - this lead me to some interesting links and info.  The direct answer is in the Alien Species Wiki.  In main Xenomorph article there's enough to figure this out.
In the Biology Section it points out that the "tongue mouth" or secondary jaws (or mouth) is what the Xenomorph eats with.  (So my original guess that the secondary one would hold the food is wrong - the large mouth holds it if needed.)  Even more, the Secondary Jaws can be used as a close-combat weapon.  It can be pushed out of the primary mouth or jaws with such force it can shatter skulls of some species.
What surprised me is that we have a species on Earth with something similar - the Moray Eel also has a Pharyngeal Jaw.  It can't easily swallow, so it grabs prey and the smaller internal jaws, the pharyngeal jaw, then bit and pull the food into the gullet.

