Previously, travelers arriving to Turkey would have to hand over $20 in cash at the airport to receive a "visa on arrival," aka a tiny sticker on a passport page complimented by the arrival stamp.
The new process will still cost $20, but that payment and the basic information needed must be submitted online. Then simply print the confirmation page and present it at immigration when arriving to Turkey. The e-Visa is good for multiple visits for a total of 180 days, but an individual visit cannot exceed 90 days.
Payment is only accepted by debit card, or Visa and MasterCard, but they're working to add more payment card options.
[UPDATE: A transitional period will bring kiosks to Turkey airports, so that travelers may obtain the e-Visas on-arrival until the news sink in enough that the entire process goes online]
Specific information for travelers from countries other than the US can be found on the official Republic of Turkey Ministry of Foreign Affairs website.