In terms of supply voltage, look at the data sheet for the device. It will tell you what operating voltage to use. Using 5V on a 3.3V device likely will damage the 3.3V device. Using 3.3V on a 5V device may not cause damage but the device may not work.
In terms of input and output voltages, an input on a 5V device will usually recognize a 3.3V signal as a valid HIGH. Do not apply 5V to a 3.3V device input unless the data sheet for the device clearly states that the 3.3V input is 5V tolerant. Otherwise a level shifter of some sort will be required between the 5V signal and 3.3V device input.
Do you have a specific case that you are working with?