IRS is an acronym for Independent Rear Suspension. An IRS diff is sort of hinged at either side of the diff centre, so either wheel can move independently of the other, one can be up and one can be down, for example. IRS is arguably better for a circuit car/cornering.
Live axle, or solid axle/rear is where the diff is one solid piece the whole length. It's better for quarter mile or straight line racing.
You can see which type of diff you have quite easily just by sticking your head under the back of your car, as the two diffs look very different. If you're still not sure, put a jack under the centre of the diff and you will see the whole diff lift if it's solid, or the centre will lift and the wheels will stay down if it's IRS.
Diagrams just to help
Solid/Live Axle:
IRS:
Hope that helps, feel free to ask any other questions