1. "Hold on" is more commonly used when doing something. For instance, if you are fixing or building something with your brother, and he starts putting something together which you shouldn't be doing just yet, you can say "Hold on. We have to put these parts together first." He would understand by this that what he was doing shouldn't be put together just yet.
2. Above when I was talking about the use of "Hold up", I also mentioned that "Hold on" could be used in the second example. However, I will explain why this is not common. Usually, in these types of situations, you would just say "Wait for me!" and this is a lot more "light and easy going". "Hold on" is more common in formal, serious situations. For example, if you are a child and you are eating dinner with your family, but want to get up and go to your room, your mom might say, "Hold on, you're not going anywhere until you finish your food!" or she could say, "Hold up! Where do you think you're going?"
"Hold on" can also be used in arguments as an expression of realization, shock, or outrage. In an argument someone might say, "Hold on. When did I ever say that I didn't want to go to your party?!" or something like "Hold on! It was you who broke my vase!?!"
However, whichever one you use, people will probably understand what you are trying to say. :) So don't worry about it too much, even though it is good to know the common uses of these words in social situation.