You can probably do a search for this, but insomnia has me so I’ll help you out.
First, wrapping well takes practice, so ask someone to show you, for sure. Bad wraps can lead to injuries if they are too tight or too loose, although mostly they are just ineffective!
No Bows are generally used for standing wraps, in place of quilted cottons, because it is much harder to wrap too tight in No Bows than regular standing quilts. Wrapping too tight can cause a bandage bow. This is not a real bowed tendon, and it won’t do much more than superficial damage, but it looks like one!
Most people use standing wraps after a hard jump school if the horse is staying in, to prevent stocking up and give support. I usually put liniment on (an arnica one I make myself or Vetrolin) underneath and leave the wraps on overnight. I’d rather my horses go out after jumping, but if they are going to be in their stall, or we are at a show, wrapping is a good choice.
Some horses get wrapped if they are prone to stocking up when in their stall, whether they jump or not. I had a horse who wore Back on Track leg wraps on his hind legs at night if he had to be inside longer than normal due to weather because his back legs would get fat without them. Standings are also used if the horse needs a sweat or poultice and/or has some sort of injury to its lower leg.
Pillow wraps are generally for shipping, instead of boots. I prefer them, and use big, thick pillow cottons and non-stretch flannel wraps for shipping. The difference between standing cottons and pillow cottons is just that the pillows are thicker. I also get them longer than standings so they can go over the ankles and up the leg a bit higher. Standing cottons are usually just covering the cannon bone.
I hope that helps!