Since I've seen this topic brought up in questions before, I wanted to present a more thorough writing on it.
First, I'll define an inverter as a device that converts a range of DC Input Power to an AC Output, usually at 120 VAC, 60 Hz but there are 240 VAC, 50 Hz output inverters for Europe and other parts of the world. The inverter can have a sine wave or a modified sine wave output waveform. The modified sine wave inverters are really just a square wave with a dead band centered where the zero crossings would be if the waveform were a sine wave.
On the subject of Sine Wave vs Modified Sine outputs, both will power nearly any AC load within the capability (size) of the inverter. However, some loads, particularly inductive loads like compressors, fans, and transformers, will run a bit hotter on a modified sine waveform.
One of the biggest problems users run into with inverters is not sizing the inverter properly for the load. This is not as simple as it sounds. The user must consider both the output wattage or current under both continuous and under surge or startup conditions. Many loads including PCs, Refrigerators, and old-fashioned light bulbs, have a considerable startup surge current which can be anywhere from 15% to 400% higher than the continuous operating current. This means, that the inverter needs to be capable of outputting this surge power for the duration of the surge which is typically from a few hundred milliseconds to several seconds. Often, a user will buy a 500 watt inverter for a 400 watt refrigerator load for example. However, the fridge might need a 600 watt capability (150% surge) to startup. Therefore, though it seems at first glance that the inverter is sized plenty large enough, in fact, it will not start the fridge in this case.
Another issue in sizing inverters is Reactive Loading. This discussion is irrespective of surge loading and any surge loading must be added to the inverter's output rating in addition to the resistive and reactive continuous loading. Reactive loads do not actually dissipate power. Instead, they draw and momentarily store AC power, then return it back to the AC powerline. However, the current that's drawn by a reactive load is real and an inverter needs to be sized to handle it. The power rating of an inverter is based on a Resistive Load for which there is no reactive power. But many loads are a combination of resistive and reactive power. To properly size an inverter for a reactive load, it's easiest to use the load's maximum current rating, normally listed on the load's identification plate or label. Fortunately, the load's rated current includes both resistive and reactive current, which makes things simple. For the inverter, the size is typically specified in watts. To compute the maximum output current an inverter can provide, simply compute the following: CURRENTout = POWERout/VACout. For example, if the inverter is rated at 2,500 watts at 120 VAC RMS, the maximum output current would be 2,500/120 = 20.83 Amps RMS AC.
Assuming the inverter has no output surge capability (most do not), the computed current rating is all that the inverter can give. Therefore, if your load has surge rating and a continuous rating, the surge rating must be used when sizing the inverter.
Finally, some inverters are rated to run continuously and some are rated for maximum output at a certain duty cycle. If the inverter chosen has an output duty cycle rating, it means the inverter must be shut down to cool after running for a while. For example, if an inverter has a rated duty cycle of 25% with a maximum on-time of 1 hour, it means the inverter must be shut down to cool for 3 hours after running for 1 hour. Some inexpensive inverters are duty cycle limited but do not say so in their specifications. Others may be rated for maximum output continuously at 25 degrees C, with an output power derating above that temperature. But since any inverter will heat-up, this is not a very practical rating. In other words, the inverter probably could only be run at full output power continuously if the user provided sufficient fan cooling to keep the inverter from going above the rated temperature. Otherwise, the user would need to reduce the load or only operate the inverter for a relatively short period of time.
These are the important things to consider when selecting an inverter. Please take care to ensure you're not selecting an inverter that won't do the job for you.