I've talked about this a few times here so pardon me if I just try and give the quick and dirty version
When you have a UPS, you are taking AC and converting it to DC (to charge the batteries). When you switch to battery power that DC is going to AC through an inverter. There are line-interactive UPS' (most UPS are of this type). Here the load runs on the main power feed. When the main power feed goes out of spec (brown out, voltage spike, or total blackout), then the UPS kicks on to battery power to drive the load (in these situations there is a slight delay in kick on that some computer power supplies can handle and other's can't). This duration is nanoseconds in length, and the ability for equipment to survive this essentially comes down to how much reserve power the capacitors can store in the power supply, and the power supplies' ability to handle a sag in power (if the capacitance is low).
The alternative to this is an Online UPS, which constantly takes the incoming power, changes it to DC, and then back to AC. This ensures that the power coming from the UPS is as clean as possible. Most business oriented UPS' are of this variety and start at a fairly high cost due to the better electronics inside, as most consumer grade UPS systems do not have the proper heatsinks and cooling systems in place to run for any longer than their batteries will last.
Of these two types, you also have Modified sine wave UPS, and Pure Sine Wave UPS (though to my knowledge, there are no non-Pure sine wave, Online UPS, as there is no point in such a cost-cutting measure in an already expensive UPS.)
Modified sine waves re-create an AC current from a DC current by chopping. The result, in its crudest implementation, is the square wave. See below for a good image of the two wave sources that matter here (Square and Sine).
Now most (unless the equipment is of dubious origin) UPS' are not completely square wave, but rather an acceptable mixture in between a sine wave and square wave, known as a "modified sine wave" (or AVR as Cyberpower likes to coin it). There are many ways of implementing this, but ultimately what all manufacturers attempt to do is as cheaply as possible make a square wave resemble the proper sine wave of AC current.
The problem is that equipment is designed around a pure sine wave model. Incoming power deviates from this all the time (for instance most equipment can survive a +/-3% variance on AC frequency) In these situations however, the equipment must either compensate (by bucking or boosting), or if there are no protections in place, the equipment succumbs to abnormalities (clocks that use the AC power to keep time are notorious for malfunctioning on square wave power).
In a nutshell, modified sine wave power causes inductive loads and motors to run hotter, and more inefficiently. Depending on the sensitivity of the equipment, the type of load, and how "modified" the modified since wave UPS is, you can expect a 20-30% decrease in efficiency when a load is driven off of a modified sine wave. The equipment either has the power envelope to spare and will run fine, albeit hotter, or it will not have the room, and will refuse to run (shut down), or the power supply will simply release the magic smoke (some plasma TV's were known for this issue).
Also, as said by javier_machuk, when you have a modified sine wave load, the Power Factor sharply decreases, this is why your load has to fit within not only the Watts load, but also the VA rating.
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