The NASDAQ market is a virtual exchange, also known as an OTC (over the counter) market. There is no trading floor, no specialist, and no central location. Instead, all the trading takes place via a computerized network of dealers. This minimizes the possibility of computers going down, since the networks are independent (if one goes down, others pick up the slack).
The NASDAQ was previously considered inferior to the NYSE, but since the technology boom, when many companies such as Microsoft and Intel grew tremendously, the NASDAQ has come to be regarded as the primary competitor to the NYSE, both in terms of prestige and trading volume.
Brokerages that participate in the NASDAQ system act as market makers. Unlike the specialists on the NYSE, market makers only control their own bid and ask quotes versus the entire trading for a particular stock. There can be many market makers in any particular stock, all trading at the same time.