An elliptical mirror is one that is designed in an elliptical shape --> which is a conic section, or closed curve - like a squished circle. Ellipses have the special property of 2 very elegant focus points, which would make it a lot easier to trap light in a region at the same time as having a place to insert it from.
It would certainly need to be in a vacuum to remain for long periods of time (so as to prevent the light from interacting / getting absorbed by particles in the air).
Mirrors can't reflect all light anywhere close to 100% efficiently, but to my knowledge, when created for a particular frequency (or small frequency range), mirrors can be made to essentially (though still not quite) 100% efficiency.
The general conclusion is that keeping light bouncing back and forth in any small, defined region, indefinitely, would be impossible; but with a lot of trouble you can keep it contained for increasing periods of time.