For using an inverter, what determines the power usage?
Your assumption is true, power is conserved if you have a 'magical' inverter that converts energy at 100% efficiency.

For a real world inverter however, you need an efficency rating, usually 80-95% but could be lower. For a real world inverter the power out can only be lower than the power in, and if your doing any kind of sizing its better to estimate the efficiency with a lower number.

$$ P_{in}*Efficency = P_{out} $$