They're called 'virtual' airlines. The story is that you create an airline with routes, website, and a fleet. Your operation is run with a flight simulation program, generally Microsoft Flight Simulator, and you fly the routes using the program.
You get aircraft painted in your color scheme, then you recruit pilots to fly with your airline. Each week, the pilots recieve assignments for which routes they are to fly, and they do so, filing a PIREP (pilot report) on the flight after they're done. The process repeats over and over again as each week goes by, and you have your airline.
Or so the theory goes. Often, it doesn't work out very well, which is why virtual airlines almost always go bust within two years, and usually within the first 3-6 months.
There is an excellent website run by a guy named Gary Kulash (who is arguably a large reason why the VA hobby is so incredibly popular today) called the VANF. It's a news source for virtual airlines, and is a good place for newcomers to the hobby to go to in order to figure out which companies are hot and which are not. It can be accessed at http://www.vanf.com.
Also, VirtualAirlines.com is the premiere site for virtual airlines, and hosts most of the reputable ones in the hobby on its site. Check it out if you're interested in virtual airlines.