Julienne basically means long or short thin strips. Long ones are nice cooked in noodle salads, Chinese and Japanese soups, or raw on top of these for garnish (especially spring onions and carrots). Short ones are good in pasta sauces, or creamy sauces for pies, though this is an awful lot of hassle.
This isn't something you necessarily want to do if you're a cookery novice. It requires a sharp knife, a decent degree of control over the knife, confidence, and patience. The results are definitely worth the trouble, though - have you ever experimented with cutting a vegetable or a fruit in different ways and seeing how different it tastes and feels in your mouth? Try comparing grated carrot or apple to large chunks of the same, for example. Start off doing julienne for garnish, and try recipes which use larger quantities as you improve your technique. Use a sharp knife and take your time!
Peel carrots, then slice into cylinders of the length you want your strips. It helps if you make the cylinders roughly the same width all the way down - eg, if your carrot changes diameter abruptly, cut there. Now, slice the cylinders lengthways into very thin rectangles - don't worry too much about keeping them stacked up. Finally, stack up a few rectangles, and slice lengthways into fine strips. Don't try to have too big a pile at once, and make sure you have a flat rectangle (ie not one that came from the outside of the carrot) on the bottom.
While you're slicing the cylinders and rectangles into smaller pieces, keep your fingers on top of the vegetable, to the left of the knife (if you're right-handed), and move them away from the knife/push the pile as a whole towards the knife slowly as you slice.
Slice off the stalk and julienne separately. Place the mushroom on the board cap-upwards and slice downwards, thinly. Keeping the slices together, take half the mushroom at a time and place cut-side down before repeating the slicing. Mushrooms are fairly easy to julienne as long as you don't treat them too roughly (the closed-cap ones tend to fall into smaller pieces if you do).
Cut the pepper into quarters from stem to base. Deseed, and remove the stalk and white membranes. The trick to julienne peppers is to slice midway between the outer skin and the inner boundary before beginning - to do this, you need small enough pieces of pepper that you can press them absolutely flat (skin side down) to the board and slice with the knife parallel to the board all the time. For a small pepper, quarters should be fine, but for bigger ones you will need up to eighths.
Once you have cut a piece, lay the halves one on top of the other as they originally came, and just slice thinly in one direction.
For the green bits, just wing it. For the white bits at the bottom, the best way I've found is to slice in half lengthways, then lay cut-side downwards on the chopping board, and slice lengthways from there.