With out seeing a circuit diagram or the push button it will be hard to give an accurate answer.
The push button is most likely a normally open type.
That means that pushing the button turns on electricity only as long as the button is held down.
This is the kind of push button used on door bells.
If that is the type of push button you have, it can be used to latch on a relay which then turns on the lamp.
I have no idea how the latch circuit of the relay would be broken to turn the lamp off.
Or it could provide a momentary pulse to a circuit board that will operate the lamp.
One push of the button turns on the lamp.
The next push, turns off the lamp.
If the push button is a Maintained type, the first time you press the button, it turns on electricity and maintains the connection, keeping the lamp turned on.
The second time you press the button, it turns off the electricity and the lamp turns off.
If you added Additional Details, like a link to a picture, we could give a better answer.
Or click my Avatar picture and email me.