What's the name for a switch that opens its contacts when pressed?

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That's called a normally closed pushbutton.

Most pushbuttons are open when left alone, and pushing them closes the electrical connection. These are called normally open. Normally closed pushbuttons do exist, but will be harder to find and more expensive when you do.

Nowadays, pushbuttons are usually just inputs to microcontrollers, so their polarity doesn't matter to their function. The micro can tell whether the button is pressed or released at any time, and then take the appropriate action.

For example, you could use a normally open pushbutton into a micro, which can then interrupt power to something else when the button is pressed. Even without a micro, you could use a relay that allows power to flow when not energized. Pushing a normally open pushbutton would energize the relay, which would interrupt power. This has the advantage of the power and possibly high voltage not being near the pushbutton where you have to worry about proper isolation. The pushbutton also need not be rated for the full power current. It only needs to handle enough voltage and current to energize the relay.

Such a switch is called normally closed. Also, pushbutton is a better word than switch, because a switch will remain in its last positions whereas a pushbutton will return to its original position once you release it.

This normally closed pushbutton opens its contacts once you push it. It makes a contact again once you release it.

The switches you consider "normal" are called normally open and make a contact once you push them.