To answer in the language of the question asked:
- Medium holes with a cutting edge on bottom (look like large raindrops) : used for shredding.
- Small holes with cutting edge on bottom : used for shredding when you want it finer than the larger size.
- Outward protruding holes with spiky edges on all sides : used for grating.
- Wide holes that look like a smiley : used to slice things to an even thickness.
As for recipes ... I generally don't use the slicing portion too often, as I can get fairly consistent slices with a knife, but if you're not so skilled, it could be used to slice potatoes for an au gratin (might need to slice them in half first, is the slots aren't wide enough to fit the whole potato), or to slice firmer cheeses. It can also be used for slicing cabbage for coleslaw (again, once cut down to fit), cucumbers or carrots for salads, etc. As the blades aren't razor sharp, there are some softer items (eg, tomatoes) that it just won't work for, that you'd have to do by hand or get a much more expensive mandoline.
The grating side I only generally use for hard cheeses, when I need a more powdery texture than I can get with the shredding sides. I've also used it for zesting citrus (although it only works in large quantities, as you end up losing about 1/2 a lemon to the groves and it doesn't release without a brush**), and I've used it for pulping carrots (was mixing them into a sauce, and I didn't want identifiable bits)
The shredding sides are the ones I use the most, with the choice of side dependent upon what size I want the resultant shreds. I use it for firmer vegetables and fruits that I'm baking into breads (zucchini, carrots, apples), potatoes for hash browns, medium cheeses for firmer cheeses (when I want this texture), etc.
Although a food processor can be used for this task, you have to consider a couple things (besides initial cost & space it takes up) : it's really easy to go and shred lots of cheddar cheese in a food processor, but it cramming all in there and fusing back together from the force and heat, defeating the purpose. Chilling the cheese helps, but you also need to keep dumping the work bowl out. If you have a strong arm and a small food processor, a hand shredder might actually be faster and give better results.
When using a shredding disk for firmer things, you do more damage than by hand, resulting in lots of liquid being given off. Sometimes, this is better ... I have a potato kugel recipe that comes out very light as I can get more of the liquid out after having gone through the food processor ... so you may want to try both tools (if you already own them both) to see which one gives you better results.
** ... I've heard you can wrap it in plastic wrap, use the grater, and then pull the wrap off to get it to release; I've never tried it, in part because I now have a microplane grater, but also because I'd be afraid of getting bits of plastic in the food. I also want to say that I saw the trick the context of pulping ginger or garlic, but it's been long enough ago that I don't know that I can trust my memory