I am not a mechanical engineer. I am trying to figure out the easiest way to design a mechanism such that, when I press down one end, the opposite end goes down as well. I'd like to look at existing designs, but I need to know the name of the mechanism in order to look it up.

The design I came up with uses two levers. When I push down on lever 1, it lifts the opposite end. This lifts one end of lever 2, lowering its opposite end. There may be other or better ways to do this,.

I found "compound levers", but those aren't quite what I was looking for.

 

Two levers as you've described is one way to do it. If the two points aren't two far apart, a linear bearing setup might do it. A push-pull cable in an "S" shape could do it, as could a setup of cables and pulleys. A rack gear driving a pinion gear, which in turn drives a shaft, driving a gearing the other side which moves a corresponding rack, is another possibility. Then there are hydraulics, servomotors and linear encoders, selsyn motors, and, and, and...