I have had trouble with nosebleeds since I was born (I'm 31). My father had the same problem and eventually had to have his nose cauterized. I would wake up in the mornings covered in blood (scared my Mom a bit...) and was always that kid in the back of the room trying to get her nose to calm down enough to learn. As an adult, I've found that as my immune system is stronger, I have less nosebleeds. And when I do have one, it's usually the first warning sign of a cold or virus that I may not be able to shake.
Just recently, a friend told me that her son experiences nosebleeds, too. Their doctor told her that her son has a superfluous vein in his nose; it's basically too close to the surface and will bleed at the slightest bump or provocation. His nosebleeds seem to usually be on the same side, as do mine. I think this may be my issue as well. It might be worth checking into for some of the folks who have posted that their kids/selves/DH are having recurring nosebleeds.
The best way to stop a nosebleed that I use is a technique my aunt taught me ages ago. Using a brown paper bag, index card, or other stiff paper, tear off a piece about one inch by two inches. Fold it up into a little strip about 1 inch across and 1/4-1/2 inch wide, which should now be very stiff, but bendable at the same time. Place this inside the mouth, along the upper lip, where that little piece of skin is attached from your gums to your lip (I'm pointing, but you can't see me. Is this clear as mud?). There must be a pressure point there, because I've NEVER had a nosebleed that didn't respond. It's so effective, I carry an index card with me when I travel, and my DH knows how big of a piece to tear off now. I've never found pinching the bridge of my nose to do a thing, and leaning forward or back have the same result: blood down my throat. I spent half my life with Vaseline up my nose at night. I usually bled through it. However, blowing the clot out (um, eew!) right off the bat is pretty effective. I usually follow that with the paper in the lip trick.
Best of luck to everyone dealing with this. It's no fun at all being the one with the nosebleed, but has to be scary watching your kids bleed profusely!