An inverter needs to supply two needs  Peak or surge power, and the typical or usual power.

Surge is the maximum power that the inverter can supply, usually for only a short time (usually no longer than a second unless specified in the inverter’s specifications). Some appliances, particularly those with electric motors, need a much higher start up surge than they do when running. Pumps, compressors, air conditioners are the most common exampleanother common one is freezers and refrigerators (compressors). You want to select an inverter with a continuous rating that will handle the surge rating of your appliance so you don’t prematurely burn out the inverter. Don’t rely on the inverters surge to start your equipment because inverters don’t like to operate in their surge mode unless the manufacturer claims to have a longer surge time than normal.

Typical is what the inverter has to supply on a steady basis. This is the continuous rating. This is usually much lower than the surge. For example, this would be what a refrigerator pulls after the first few seconds it takes for the motor to start up, or what it takes to run the microwave  or what all loads combined will total up to. (see our note about appliance power and/or name tag ratings at the end of this section).
You can use the following formula to determine the size:
Volts * Amps = watts
or
Watts / Volts = amps
1250 Watt example:
1250 / 120 Vac = 10.41 amps ac (typical number found on equipment)
or
1250 / 12 Vdc = 104.1 amps dc (battery drain per hour)
Here is an example:First, you need to determine what items you need to power during a power failure and for how long. Here is a brief example (watt requirements vary):
Lights  About 200 watts
Fridge  About 1000 watts
Radio  About 50 watts
Heater  About 1000 watts
Total wattage needed is 2250 watts. The fridge and heater have a start up power requirement so let's allow 2x the continuous wattage for start up requirements. 2250 * 2 = 4500 watts
Second, select an inverter. For this example, you will need a power inverter capable of handling 4500 watts. The continuous power requirement is actually 2250 but when sizing an inverter you have to plan for the start up so the inverter can handle it.
Third, you need to decide how long you want to run 2250 watts. Let's say you would like to power these items for an 8 hour period. Well this can be tricky because heaters and fridges run intermittently. Let's assume all of the appliances will run 40% of the 8 hr period which is 3.2 hours of actual run time. We need to convert the ac watts to dc amp hours because that's how batteries are rated.
To convert ac watts to dc amps per hour you divide the watts by the DC voltage (usually 12v or 24volts). Let's use 12volts since it is the most common.
2250 watts / 12 vdc = 187.50 dc amps per hour
187.50 is now your power requirement per hour
You have now determined that 187.50 is your power requirement per hour and now you need to multiply that by total hours of run time which is 3.2 in our example.
187.50 dc amps per hour 3.2 hours = 600 dc amps
Because you are using an inverter, you want to calculate the loss for converting the power which is usually around 5%.
(600 dc amps * 5%)+ 600 dc amps = 630 dc amps per hour (this is how much power you need in an 8 hour period running your appliances 40% of the time)
Fourth, now that you know your total power requirement is 630 dc amps we can select a battery source. Most typical deep cycle batteries are 6 volts or 12 volts. I will give you two examples using each voltage.
12 volt battery example:
If you select a 12 volt battery rated at 100 dc amps you will need 6 or 7 batteries in parallel (I will explain parallel vs. series later).
630 dc amps / 100 dc amp battery = 6.3 batteries
6 volt battery example:
If you select a 6 volt battery rated at 200 dc amps you will need 6 batteries in series and parallel.
3.15 * 2 = 6.3 batteries
No, I didn't make a mistake. When you use 6 volt batteries, you have to connect them in series to reach 12 volts. Then you connect each series pair of 6 volts in parallel to create your 12 volt battery bank.
What is series and parallel you ask?
When you connect batteries is parallel you are increasing amps. When you connect batteries in series you increase voltage. In the battery world, it is better to limit your parallel strings. It is better for your power system. In this example, I would recommend using 6 volt batteries because of the number of batteries this example requires.
How do we charge these batteries? You will need a charger to charge the batteries when you have access to city power. Most deep cycle batteries need a "smart" charger so the charger doesn't damage the batteries. In this example, you will need at least a 40 amp charger if not bigger. The bigger the charger, the faster the charge. Make sure your charger is for 12 volt batteries because the system we just identified is a 12 volt system.
You will also need cables. For this example, a 4 AWT (0000) cable is required to handle 4500 watts of start up power. That is huge cable. You may also want to consider an inline fuse. A 500 amp for this example is perfect. To figure out the size of fuse you divide your ac watts (start up) by dc voltage.
4500 watts / 12 vdc = 375 amps
You would need a 375 amp fuse or bigger. I recommend a 500 amp just incase you were to max out the 5000 watt inverter.
This is just a brief example. There are many different ways to set up your system. You can use solar panels, wind etc.