What type of cloud is this? I live in a sub-tropical region of Australia.

Earlier today, we experienced temperatures of 31C and it was clear about 2 hours before I went out and noticed the following clouds directly above my house - the entire sky either has this or is very grey.

There is a constant background of thunder and distant lightning is observed - the wind is picking up a bit, though still a breeze. No rain at this stage.
These are mammatus clouds. The signify instability (via negative buoyancy, convective mixing, entrainment or other sources) at the cloud base which causes the bulbous udder-like structure. As in your case they can often be observed in association with strong convection.