You guessed it. We are talking about an AR15 rifle (or an AR10 or AR-15). 
Direct Gas Impingement means that gas from a fired cartridge is send directly into the bolt carrier group (BCG). This gas pushes the bolt forward (in the BCG against the barrel extension) which in turn pushes the BCG rearward against the buffer and buffer spring.

Now even though it's called direct gas impingement, the bolt is acting as a piston inside the BCG. The bolt has gas rings to keep the gas inside the BCG until the bolt gas rings travels past the exhaust holes in the side of the BCG. By definition, "direct" impingement means the gas works directly on the object it has to move (the BCG). If you consider the BCG as a unit, this is certainly true but the BCG does contain a piston (the bolt) so it's not 100% accurate. There are guns that operate on a truly DGI where gas is directed into a chamber and the gas directly interacts with the bolt or action.

The beauty of the DGI system is it's simplicity and it has many advantages like: inexpensive, standardized mil spec parts, reliability, parts availability. Because hot gas is sent directly into the action, heat and cleaning can be an issue especially if you're using a suppressor.

In a gas piston system, the gas operates a piston running parallel to the barrel and using a rod, pushes the BCG directly rearward. No hot gas is introduced into the BCG or chamber directly. The advantages are the bolt stays cooler and cleaner. Some disadvantages are cost and interchangeability of parts. Some opinions are that this system is less accurate but testing has proven otherwise. Another issue was found when adapting existing bolts and BCGs to operate with a gas piston was that the force exerted by the piston is off center to the BCG (on top) and this can cause excessive wear and failures.

If there was there was an industry standard for a piston design the costs would certainly come down but because of the vast amount of manufactures of the DGI systems, this may never happen.

Here's a great video of both systems here. (Linking this video does not constitute and endorsement of their products.)