I am traveling to Jamaica where the drinking age is 18. I am 20 and I plan on drinking. If I show my American driver's license will I be given a drink or will I be turned down?
You can use your passport. It is a document which is recognized internationally. Try your license first, though.
You can also get an international driver's license. In the US, for example, you can get one for a fee at your local AAA office. This is a photo document, recognized many countries (you must present it along with your regular driver license). Additional: as long as you are not intoxicated when you land back in the US, you are fine.
Jamaica is not a strict enforcer of drinking laws, so you may not even be asked for ID if you don't look obviously much younger than 18. If they do ask for ID it is up to the establishment what they accept.
Your US drivers license is likely to be accepted (there are no rules about not accepting foreign documents like there sometimes are in the US). If your US license isn't accepted then your passport is almost certain to be.
And just in case you were asking, identifying yourself as American will not prevent you from drinking at age 18.