Is your tank a saltwater tank? Live rock is usually rock found with living things in it and on it in salt water.
If your tank is fresh water, your rock in your tank is probably just turning brown from diatoms. This is normal in a newly established tank, and will take about 2 months for it to disappear. It's sometimes called brown algae, but it's not a true algae.
No.... liverock is rock that has broken off of coral reefs in hurricanes and such.... It has plenty of nice living organisms on it and is used as biological filtration and eye candy in Saltwater tanks... it ranges from $2-$7 a pound and you need at least a pound per gallon of water... so people with big tanks better be ready to shell out big bucks.