A power supply lists the voltage it produces and the amperage it is capable of producing (and safely sustaining).
A load (device which uses the supply) lists the voltage it needs and the amperes it will draw.
So if your device says "12V 1A" but your supply says "12V 0.1A" then the supply cannot power the device. But if the supply says "12V 1A" or "12V 100A" (or any value over the rated amperage, really) then it should be fine since, as you mentioned, the device will only draw the amperes it needs (assuming that the voltage on the device matches the voltage on the supply).
Does that make sense?
Regards,
Torben