To julienne a vegetable or other ingredient is to cut it into long thin matchstick-sized strips 1/16th to 1/8th of an inch wide and 2-3 inches long. A baton is larger, almost 1/4 inch wide.
To julienne a carrot, first cut it into cylindrical sections of the desired length. Trim each section so that the sides are straight and right-angled (almost like a cuboid). Slice into flat panels of desired width. Stack these panels and slice evenly into matchsticks.
Julienne is also a name given to a consomme garnished with julienned vegetables.