Lighting, Light Fixtures, Ceiling and Exhaust Fans - no ground in light switch

I wanted to install a dimmer switch for the overhead light in the dining room. The instructions state to connect the ground wire to a certain screw, but this doesn't resemble the wires I have available.

I have two black wires, both of which were connected to each end of my old switch. I have connected them to the appropriate place on the dimmer.

I have two white wires that are connected together with wire nuts, and I have two copper wires that are connected together with wire nuts.

If no ground wire was connected to my old switch, then is it ok for me to not have a ground on my new one, or will my house explode?

thanks,
Tony

I'm not sure whether you meant the ground wire on the dimmer to some screw elsewhere, or a ground wire elsewhere to a screw on the dimmer. If the former, connect the ground wire on the dimmer to the two bare wires in the box. If the latter, then add a bare of green wire (of the same size as the existing wires) to the wire nut holding the two bare wires together, and connect the other end of that wire to the "certain screw" (presumably a green one) on the dimmer.