@Linkat,
The process for making regular and Greek yogurt starts off the same: milk is first heated, then cooled to the desired fermentation temperature (106-114°F) before bacterial cultures are added. The mixture is then left to ferment until the bacteria grows, produces lactic acid, and gels the milk proteins to produce regular yogurt.
To make Greek yogurt, regular yogurt is strained extensively to remove liquid whey and lactose, leaving behind a thicker-textured yogurt.