Open ground, as I initially guessed.

The button isn't useful unless you're testing a GFCI outlet, so just ignore it.

I'm no electrician, but as I understand it, you'll need to run a ground wire from each outlet that you want to be properly grounded to your electrical panel. Then inside the panel, If I understand correctly, the ground bus should be bonded to the neutral bus, and also connected via a grounding wire to a 8 ft copper rod that's driven into the ground. Any electrician should know what to do and would be incredibly cheap by US standards. The hairy part is what's involved in running that wire from point A to point B.

Also, any outlet that you don't want to run a ground wire for should be switched back to a two prong outlet or labeled so that people don't mistake it for a grounded outlet.