Watts are like speed, not distance.
When your power consumption meter says "70W" that is a rate, like your car saying "70mph" on the speedometer. To calculate how much distance you covered, you might say "I drove 70mph for 6 hours = 420 miles of distance."
In the same way, you need the time variable to calculate how much total power (distance) you used. Usually that's expressed as Watt-hours (Wh), so you might say "I used 70W for 6 hours, resulting in a total of 420Wh."
Personally, I find it more useful to think of it in Amps (current) and Amp-hours (Ah) because that's how batteries are usually rated.
If your meter reads in Watts, you only need to know the voltage to figure out the Amps. The formula is simple:
A = W / V
So if we're talking about your 12V system:
70W / 12V = 5.83A
If that is a steady draw, this makes it easy to calculate how long you can run something on a battery. Once again, think of Amps as speed and not distance.
For that analogy, we could imagine that you want to know how many hours it will take you to cover 100 miles at a speed of 5.83mph. Simple division tells us that it would take 17.15 hours.
Let's apply that to your situation and say your battery has a capacity of 100Ah. At a draw of 5.83A, your battery would last for 17.15 hours.
In reality, you can't discharge your battery completely. Usually 50% is safe. So really you could run that 5.83A draw for about 8.5 hours before you needed to recharge your battery.
If we're talking about your 110V system, the calculations don't change. Why? Because your 70W meter rating is already the product of Amps multiplied by Voltage. The A = W / V formula can be reconfigured in two other ways:
W = A × V
V = W / A
A simple mathematical proof can verify that 70W on the 110V AC system is 70W coming from the battery (with rounding):
110V × 0.64A = 70W = 12V × 5.83A
Ignoring inefficiency from the inverter that is producing the 110V AC power, we know the system is drawing 70W from your battery and your battery is 12V, so the calculations remain the same.
Hopefully this helped clear things up. I think people get confused because of the time variable, as they don't realize that units like watts and amps are more like speed than distance.
Both Watts and Amps are measures of rate (like speed), while Watt-hours (Wh) and Amp-hours (Ah) are amounts derived from the rate over time (hours).
5 Amps is a rate, but 20 Amp-hours is an amount (5 Amps drawn for 4 hours).
100 Watts is a rate that takes voltage into account (Amps multiplied by Voltage), while 400 Watt-hours is an amount (100 Watts drawn for 4 hours).