This guide discusses some of the stuff one should consider about power inverter designed to convert DC (direct current) from batteries into AC (alternating current) for running normal appliances.  Most common these days are inverters which convert 12 Volts DC into 120VAC, they are widely available for off grid cabins, RV's, trucks etc.  Nicer, more expensive units are usually required for off grid homes.  Some of these inverters can actually be 'grid tied' to allow for selling extra power back to the power grid. 

Most important considerations

Rated output and surge capacity
Inverters are available from about 75 Watt up.  If your only goal is to charge your laptop off your car battery then a very small/inexpensive inverter would probably do the job.  If you expect to run skill saws and other power tools then you might need something a bit larger.  As you would expect, the larger the rated output the higher the cost.  Most inverters have a rated output (an amount of power they can convert constantly) and a surge capacity (a higher amount of power they can produce for a limited amount of time).  Most inexpensive inverters are light weight and they have small transformers which heat up quickly if over loaded - these will have less surge capacity (they may have more trouble starting large electric motors and such).  More expensive inverters often have large transformers and they do much better at handling very large loads for small bits of time.  These units can also use their large transformers backwards as battery chargers which is a nice feature.

Wave Form
This is an important one!  Power from the grid or a generator comes in the form of a sine wave and this is what almost all appliances are designed for.  An alternator creates a sine wave easily - it's more complex for an inverter though.  The closer they get to a real sine wave the more complex and expensive they become.  Inverters are designed with the compromise between cost - and quality (as is everything I suppose) in mind - the better the wave form the higher the cost.

square wave

Pictured above is a 'square wave'.  Square wave inverters were made for years - until perhaps 25 years ago they were the norm.  Many normal appliances would work OK off them - although some motors were not very efficient - audio equipment would hum badly, televisions would sometimes have interference on the screen etc.  I don't believe anyone makes these anymore - they are obsolete for all practical purposes.

Modified sine wave

Even though they call it a 'modified' sine wave' (It's all about marketing) - it's really more of a modified square wave.  In the picture above you can see the 'modified sine wave' imposed over a real sine wave.  In the eyes of most appliances it looks a lot more like a sine wave than a square wave anyhow!  Most stuff runs ok from a modified sine wave inverter but there are plenty of exceptions such as some microwave ovens, some small battery chargers (like cordless drill chargers) - and AM radio will probably not work.  Some cordless drill chargers can be destroyed by modified sine wave output so it's something to keep in mind.  Modified sine wave inverters are most common and can be had at very low cost these days.  They are ok for some applications but if you are living every day with your inverter you may want to get a pure sine wave inverter.

Pure sine wave

No inverter can make a perfect sine wave, but with lots and lots of little steps they can produce something that is for all practical purposes the same - and in the eyes of your appliances their output is every bit as good, or better than grid power or power from a generator.  Pure sine wave inverters tend to cost more, and they tend to draw more current when idling.  This is often made up for by the fact that some loads will run more efficiently off them.

As most things in life, you get what you pay for so please consider carefully.