The problem is your inverter is going to have a hard time recharging the batteries but I'll post the math and explain as best I can:
First of all to determine the Amp hour rating, check the battery, it should be physically written on the battery somewhere, ALSO make SURE it is a deep cycle battery.
As for the math it's quite simple:
The total amount of power a battery can store is:
Amp Hour Rating x Battery Voltage = Watt Hour Rating.
So lets say you have a 100Ah (Ah = amp hour) 12v battery, that means it stores 1200watt of power for one hour. (12x100 = 1200).
So from there to calculate the actual time your appliances will run you take the total LOAD that you will be running, for example a TV of say 300w and a computer of 400W. That is a total load of 700W.
So your single 100Ah 12v battery will last 1200/700 = 1.7 hours.
If you connect many batteries in series the voltage increases so if you connect 2x 100Ah 12v batteries you have 100Ah 24v which is 2400W which would give you 3.4 hours on the load above. ONLY connect the batteries in series if the Inverter supports 24v else
Parallel, which for 2x 100Ah 12v batteries would give you 200Ah 12v which also translates to 2400W. You can find many source on the inter that explain how to connect batteries, circuits, etc. in series/parallel
As for the Amp rating of the Inverter they obviously don't know what they were talking about because a 12amp inverter could mean anything (12amps on which voltage?).
Also disregard the VA rating Volt Ampere and Wattage may seem the same (indeed because watt = Volts x Amps) but the VA rating is what is usually used to rate a transformer, so basically the transformer (and perhaps certain other parts of the circuit) can handle a load of 1000VA but the total load is still restricted to 600W.
Also on the calculations above, a electronic circuit is never 100% efficient so best to assume 5-10% loss.
EDIT: Also remember that, best case scenario, if the inverter is running of the wall socket it can recharge the battery at full load which is 600W but chargers are grossly inefficient so it'll probably take 3-4 times as long to recharge VS. the amount of power you got out of it.