1000 watts divided by 220 volts equals 4.54 amps. So you would use 4.54 amp hours every hour it was on, or 45.4 amp hours over 10 hours.
Of course, you will need an inverter to change 12 volts dc to 220 volts ac, and there will be some losses there. The best inverters are about 90% efficient, so figure 5 amps an hour and 50 to 60 amp hours over 10 hours.
You didn't say exactly what the appliance is, but be aware that if there is a motor in it, it will need a heavy starting current to overcome the inertia and get it spinning before it settles down to it's average running current. Typically, starting current can be 4 or 5 times the running current, so you will need to size your inverter with that in mind - you might need a 2 or even a 3 thousand watt inverter to run your 1000 watt device. Be aware that 220 volt inverters of that size are not cheap. Also be aware that you will need a large and probably expensive battery charger to recharge the batteries every day.
Anyway, assuming you are not running anything else off of them, a pair of 6 volt golf cart batteries, wired in series to provide 12 volts, should do you just fine. They will give you 200 to 220 amp hours. Since you should never discharge batteries by more than 50 % for longest life, that will give you 100 to 110 USABLE amp hours in your battery bank.
Regards
John