Please clear up this confusion for me:

I just watched the video on khan academy @7:30 where the guy explains newton's 3rd law. He explains that for a box on a table, the forces equal out so it's at rest. I also understand that F=ma so an object with less mass may have more acceleration for an equal force.

But then he said if you put the table on a rocket ship, the force of gravity is not a partner force to the force moving upward. I'm having a very hard time reconcilling the terminology of what's happening here because I can calculate the force of objects moving sideways along the ground, but am getting lost somewhere in how gravity is not a partner force to the force on a body moving upward.

I'm also confused about a force "pushing down" but at the same time that means a force upward? So then is the force down or up? I know there is an equal and opposite force somewhere in this mess but also that something has to be exceeding gravity or the rocket wouldn't be able to escape?

I'm looking at other stack threads and they're saying it has something to do with different objects relative to each other but that's even more confusing.