cheap airline tickets on Tuesday midnight

As you know, airfare does depend on the airline, but it also depends on the dates that you're looking to travel on. Generally speaking, it's a bit cheaper to fly on a Tuesday, Wednesday, or Thursday. And generally speaking, the airfares you find will be lower if you search for them on a Tuesday afternoon compared to a Monday. This is because airlines often introduce new fare sales on Tuesday mornings. I've actually gotten gypped a couple of times for not following this rule (I booked a flight on Monday, then saw that the airfare dropped the following day).
the best "hour" to buy tickets is on Wednesday at midnight (midnight at your airline's base city, not necessarily your midnight). airlines release holds on tickets every day at midnight
Apparently, this is the time when unbooked fares are heavily discounted for the public and made available for a quick sale.
The best thing to do is to buy your ticket early. The longer you wait, the more expensive the airfare will most likely be.Discount fares have advance purchase requirements (usually 14 or 21 days), so there is an advantage to buying before then.
Usually, your tickets are going to be cheaper if you buy them from the airlines site directly. They don't charge service fees, so it saves you a few dollars. But it's more of the fact that the airlines are easier to work with if you book with them, I've heard a lot of bad things about booking with sites like Expedia and Orbitz. It's always worth checking directly,you have plenty of airline choices. American, Continental, Delta, United, US Airways, AirTran, Midwest/Frontier, and Southwest are the options I can think of off the top of my head. Sites such as Expedia and Orbitz will also yield comparable or equal fares to what you would find at each airline's site.