What is a Kitchen utensil used for thinly slicing vegetables?

A Mandoline. It consists of a flat "board" with a blade embedded at an angle, with the edge jutted upwards. The food is pushed along the base (often using a second piece, a hand-grip with sharp "teeth" that are pressed into the food to hold it, and protect the hand) and with each slide, a slice of the food is sliced off and dropped to the underside of the base.

The main advantage of using a mandoline is to create uniform slices, which is beneficial both for aesthetics and for uniform cooking. Using the hand-grip is seriously recommended as one slip is enough to slice of a chunk of skin.

The blade can also be supplemented/replaced with metal teeth that created long, thin strips (cutting along the vegetable's width and length at the same time). This is called julienning.