To julienne a vegetable means to cut it into small, thin pieces the size of matchsticks.
If the vegetable is to be cooked, a julienne cut allows for a vegetable to cook rapidly and evenly, and integrate well into a mix of other ingredients, for instance in a sauce. If the vegetable is being served raw, a julienne cut allows the cook to make a fine, delicate garnish for salads or as an ingredient in something like a salade chinoise, etc.
The technique is used on firm vegetables such as potato, celery, carrot, peppers, turnips, parsnips, etc. Onion is too difficult to julienne owing to its layers, and soft vegetables such as tomatoes can be done, but are difficult. Items such as citrus peel can also be julienned.
A Julienne cut ends up with vegetable pieces about 1.5 mm x 1.5 mm square and 5 cm long (1/16th inch square x 2 inches.)
To do this, the first step is to peel and wash the vegetable, then regularize its shape into a rectangle or square by topping and tailing it and squaring off the sides. Then cut it into 5 cm (2 inch) long pieces, then cut each of those pieces into 1.5 mm (1/16th inch) thick slices. Stack those slices, then cut them lengthwise into 1.5 mm (1/16th inch) wide sticks.
Note: Some chefs call the above measurements a “fine julienne”, and define a “regular julienne” as being 3 mm x 3mm (1/8th inch x 1/8th inch.) Most, though, reserve that 3 mm x 3 mm dimension for the term “allumette.” That being said, preference for exact size will vary from teacher to teacher and chef to chef.
A julienne cut can also be achieved with a mandoline.
Vegetable pieces leftover from the cut (i.e. the parts that you topped and tailed and squared away) can be used in soups or stocks, etc. The peelings can be used as stock.