What does the number of seats available in the details for a flight mean in reality?

When you search for flights you will receive back a list of available flights to meet your search request. Each flight has a tab to allow you to get more details about the flight, including connection times, baggage allowance and number of seats available for the booking class that was returned for the search. If you were looking at economy, it will show you how many seats are left in the economy cabin. 

It is NOT saying that the price returned is available for every seat. For example, there may be 100 empty seats on an economy flight, but only 3 seats left on a promotional fare of 'Z' fare basis. The system does not identify the number of 'Z' seats there are available, rather it tells you how many seats there are left in the cabin at that particular moment. In this case it would be 3 (Z fare) + 97 (all other fare basis's) for a total 100 available seats in Economy.

If you find what you need for a trip, it is best to purchase on the spot as public demand and airline passenger loads change by the second. It is better to have an airline seat than not to have an airline seat if you are on a strict schedule and/or budget.