When cooking soup, should I cover the pot or leave uncovered?

Liquids are reduced if the lid is off. Up to you..

A lot of flavour escapes in the steam. I like it better, cooked in a covered pot, in the top of a double boiler. No danger of burning, as long as you don't let the water in the bottom pot boil away.

Depends on multiple factors. If you want to reduce it to intensify flavors and thicken it up then leave the lid off. If you want it to keep its current volume then lid on. If you leave the lid off for an extended time, though, you're gonna want to make sure it doesn't completely dry out and end up burning.

When I see my mom cook, she only covers it when she simmers it. Not when it's at a full boil.

It completely depends on what you want out of your soup. Leaving the lid off will cause the soup to evaporate, leaving you with a thicker, more strongly-flavoured soup. Just uh, less of it. Leaving the soup with the lid off unattended can reduce it too much though so you want to be aware of what you're doing.

So in short - taste it and see what needs to happen to it. If you don't know, I'd lid it. You can always reduce later.

Ever hear of evaporation

I normally do mine in the microwave