Lingots are Duolingo's virtual currency. You can earn them by, for example, making exceptional comments and give them to others when they do the same. They do not cost real money. To give someone a lingot, select "Give Lingot" under a deserving comment or post that they created.
You can also use them to buy cool stuff in the store :)
Scroll down to the bottom of the store page to see other ways you can get more lingots.