Greek yogurt is more properly called "Greek-style" because it is simply yogurt that has been drained to remove more whey or thickened to resemble a drained yogurt.
Greek-style yogurt is extremely easy to make and lends a creaminess to store-brand and other inexpensive yogurts.
All you need is unthickened plain yogurt, a screen sieve or loosely woven cloth, and time.
At my local supermarket, a pint container Fage 0% Greek yogurt is $3.79 whereas a quart of the store brand plain yogurt that yields a pint of Greek yogurt is $2.39. Less than 2/3 the price! And free whey, too!
Scoop the yogurt into a screen sieve over a bowl large enough to keep the bottom of the yogurt above the resulting whey, which will total a pint or more of liquid. If you do not have a screen sieve, use a colander lined with cheesecloth or other loosely woven cloth.
Cover and place in the refrigerator to drain for six hours or more. The longer the yogurt is allowed to drain, the stiffer it will become. 24-hours yields a yogurt thick enough to spread known as labneh or yogurt cheese.
If the whey does not readily drain from the yogurt, it is likely that the yogurt was ultra-pasteurized or treated with thickeners and stabilizers. Such yogurt never will drain. Many of the nationally distributed organic yogurts will not drain whereas inexpensive, less thoroughly processed, store-brand yogurts often drain very well.
When the yogurt has drained to the thickness you prefer, refrigerate and use as you would any Greek yogurt. I like it thick enough to stand a spoon. The whey may be used for bread-making, feeding animals, smoothies, soups, etc.