What is a julienne cut?
The French have names for a lot of cuts,  not just 'dice' and 'mince'.  (and I don't mean 'large dice').  And that list is incomplete -- it's missing chiffonade ... I have no idea how many others.

Julienne is a cut where you have items looking like small sticks -- a few inches long (about 10 cm) , and about 1/8" (3mm) square.

It's also called 'matchstick cut' in English, if you don't want to use the loanword.

It's generally done with vegetables:


Peel the item if needed
Cut it down to length
Slice one side to flatten it
Place the flat side down, and slice into planks about 1/8" across.
Stack up some of the planks, and slice them every 1/8" to make sticks.


... and there are mandolines (food slicers) that can do this style of cut by just pushing the food through the blades.

As for purpose -- it's just one of many cuts that you can use to try to make sure that things are cut evenly.  (especially when you might have more than one person doing food prep).

It's useful for trying to make sure that items cook (or pickle) quickly.  It can also help make sure that harder vegetables (eg. carrots) are easier to eat (both to chew and to pick up with a fork).  It can be used to otherwise change texture in a dish.

It can also be decorative if you cut up things of a contracting color (or colors) and sprinkle them on top -- bell peppers, carrots, apple, etc.)