Live rock by definition, is porous rock with live things growing on it and critters living inside it, to varying degrees. A nice piece of live rock might have crabs, shrimp, worms, sponges, coralline algae, tunicates, coral, etc living in and on it.
Most of the LR
you encounter today is aquacultured live rock which means that it was porous rock that was dry mined and then dumped into a leased plot of ocean for a year or a couple years (enough time for stuff to "seed" the rock and fro critters to move in). The company that does this then goes back and dives down to the rock to "harvest" the seeded rock, take it back to holding tanks, and then ship to you.
Cured means that the rock is to the point where there is no die-off, so as it comes out of the ocean it is fully "cured" because everything is healthy.
After the rock gets pulled out and shipped all over the place in some cases, die-off starts to occur because some of the critters cant survive the ride.
In that case you need to "cure" the rock by placing it in a large bucket of circulating SW
and do water changes and even run a skimmer. Curing just means that you place the rock in that good SW
to sustain the life of the living guys, and let the die-off stuff wash into the water so that the rock is no longer emitting yucky dead stuff that causes ammonia spikes etc.
if you get LR
from somewhere like liverocks.com then they take great care to keep everything alive so that by the time it gets to you there is extremely minimal die off. By this I mean if ou want to you can put it right in your tank and run a skimmer to handle the little tiny ammonia spike in most instances.
Hope this helps a little.