I am trying to add a light into an existing circuit. There is a light wired to a switch, and the light is at the bottom of my basement stairs. The new light is at the top of the stairs. I was thinking I could run the wire for the new light to the existing switch, which is at the top of the stairs. When I connected it up, the light at the bottom of the stairs lights up when the switch is on, but not the new light at the top of the stairs. When I turn the switch off, the new light at the top of the stairs comes on, and the light at the bottom goes off. I am obviously doing something wrong...
Do I need to use a different kind of switch? Or maybe my wiring is incorrect?
Thanks.

Almost certainly you have to run a wire to the other light.
You cannot run one light from one switch and the other light to the other switch. It will not work.