Question about "cheap" airline tickets

I've heard that there are a number of organizations that sell airline
tickets cheaply. Is there some consumer-oriented website that rates the
cheap-ticket organizations? Can these relatively inexpensive airline
tickets be bought at the last minute or within the week before travel,
or do they require advance purchase of two or three weeks? Is it
necessary to pay to join the cheap-ticket groups? Where can I go to
learn more about these organizations? Thanks.