I'll give it a shot. Notifications is the umbrella term. There are different ways the phone can notify you of things. A notification can be a banner across the top of your phone. It can be a sound and/or vibration. It can be a little red number, called a "badge" on the apps icon. It can be all three at the same time.
Personally, I turn off sounds for most of my apps because I find them annoying. I leave it on for things like messages. I also don't like badges for most things.
You can set your notification preferences for each app individually in Settings>Notifications. I check it every few months because, sometimes, new apps trick me into turning on notifications (or I had a short-term reason for turning them on).
Here is some (a lot, actually) of additional information:
Stay up to date everywhere you go - Apple Support
It's worth reading when you are in the right mindset.