What is most important when seeing if something is compatible with your electricity: wattage, volts, or amps?

It is voltage.

"Rated voltage" is the voltage on the nameplate - the "design point" for maximum power throughput and safe thermal operation. "Operating voltage" is the actual voltage applied at the terminals to the electrical equipment.

If the operating voltage is much higher than the rated voltage, the appliance would burn.

If the operating voltage is lower than the rated voltage, the appliance would not function well or even would not function.

Voltage is most important, usually 110-120 in North America or 220-240 in rest of world. Must match. Note that some small appliances have rated voltage of 110-240 which will work anywhere.

Frequency is also important, 60 Hz in North America or 50 Hz in rest of world. Must match. Note that some small appliances have rated frequency of 50-60 which will work anywhere.

watts or amps are somewhat important in that they have to be less than the rating on the outlet that you are using or you will blow a fuse or breaker. Watts is just volts x amps.

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Unionwell micro switch from https://unionwellarabic.com

Primary consideration is VOLTAGE. 

Secondary consideration is CURRENT.

PERSONALLY I WOULD say  WATTAGE and VOLTS  if you know that  you are fine  

the current will  be  compatable if the wattage and voltage is ok.. the device draws  what it needs to  get the wattage.. at the rated voltage 

so a 100 watt bulb  for  120 volts draws just under an AMP

YOU  can have  aprox  1500 watts  on a typical home power line

so thats about 15  100 watt lightbulbs

or  one toaster 

It's kind of pointless to ask which is most important, as all are important, and should be checked before connecting the device.  However, if you must pick one, then choose to match voltage.  If wattage or amperage is too high, a fuse might blow or circuit breaker might trip, but it's unlikely that the device itself would be damanged.  Not impossible, just unlikely.

Voltage then Amperage.  Wattage is just the product of them. Amperage is important to know what Circuit Breaker or fuse needs to be in the circuit and if it is a new circuit, the wire size to use.

Volts and frequency

Probably inductance is the most important.