Invert-huh?

So what's an inverter anyway?

Your appliances run off 220V Alternating Current (AC), whereas a battery might store Direct Current (DC) at 12V.
So the inverter's job is to pump up the battery voltage to a couple hundred volts, then switch it back and forth really quickly to generate Alternating Current.

There are two types of inverters: square wave and sine wave. Square wave inverters kindof suck, so the marketing guys had a wank session and decided to call them 'modified sine wave' inverters.

What's the problem with square wave inverters?

The short answer is that, like a cyclist's legs, smooth is better.
Every time you have a quick change in voltage, a large amount of electrons will flow quickly to make the change happen.
Large amounts of current flowing then suddenly turning off is bad because it creates sudden surges. Things with motors in them especially don't like this because of things*. Computers and small electronics don't mind as much since they are converting the Alternating Current back to Direct Current anyway.

*Disclaimer: I'm dumbing this down severely, so rather learn about things like reactance, harmonics and filtering from someone better at explaining!

A sine wave is the smoothest sort of wave, but actually making a sine wave from DC is quite tricky because the electronic components that generate the wave can either switch on or off. This means they have to switch so fast that the resulting voltage averages out into a sine wave without anyone noticing. (I should probably put in another disclaimer here too.)

So long story short, pure-sine inverters cost more because they won't burn out your fridge motor.