What are villi?
Villi is the plural of the Latin word villus, which means "shaggy hair" or "tuft of hair." Thus, villi are structures in the body that look like hairs.
Villi can be found all over the body. The small intestine, for example, has a number of villi that stick out from the intestinal lining. As food moves through the small intestine, the villi slow down the food's movement and basically catch small particles of food. The villi have blood inside of them and when the villi catch the food, they absorb its nutritional elements into the blood, which then, of course, circulates around the body.
As noted by rose011295, villi also exist in the placenta, which provides nutrition for a baby growing inside the mother's body. These days, doctors can test the villi in the placenta to try to find out if the unborn baby may have certain illnesses.
Check out the links to the videos below for a visual look at villi.