As above, live rock is the porous carbonate rock that makes up reefs. It contains a variety of bacteria that live in it's numerous multisized holes and perform aerobic biofiltration near the surface of the rock and anaerobic further towards the middle. It may hopefully also be the home for sponges polyps, crabs, copepods, worms and many other creatures usually good, occasionally bad. It's usually a good source of algaes (desirable) as well.
It's the easiest way to et good filtration and is the centre of what's known as the 'Berlin system' , along with protein skimming. It's used in both reefs and Fish Only w/ Live Rock (FOWLR).
When it's transported , stuff sually dies, the rotting off and cleaning process is called curing, and should not be done in the display as it can be grim.
Many people buy their live rock over the internet, but I feel this can be a bit dodgy for the begining aquarist, especially those with a small tank. If you;re buying 50 pounds plus it's worth it BUT 1. You can't see what you're buying, 2. A lot of larger orgabisms will likely die. If you only want 10 pounds for example for your 10 gallon, accept the cost and buy what you can see. That's just my opinion, but there's a lot of good and not so good rock out there. Plus you better like the shape, aquascaping because it's not clever to keep shhifting it round.
There several grades, I went to a shop that I think is awful (their fish die too often) last week , but they had the best rock I'd ever seen, even with button polyps, blue sponges, they wanted about 22 dollars a kilo, for a small tank I'd have bought it though