LSD is short for limited slip differential and is standard on all models except base. The Z uses a viscous limited slip; there are other types as well(some are considered better).

The purpose is to lock up the differential in the case that the tires turn at different speeds; simulating a solid axle which is considered preferable for launches, stopping the one wheel burn out.

The reason its better than a solid axle is that it offers limited slip, allowing the inside wheel to turn slower in a turn. It also opens up the possibilities of independent suspension(which the 350Z has).