Variance measures the difference between the spot price over different times and the average price, and represents the average of the squared mean. Part of the reason why deviations are squared in order to calculate variance is so that they are always positive, and therefore would not cancel each other out with similar deviations in opposite directions. 

The significance of variance for regular option trading is how the square root of it is realized volatility (the expected percentage move over a period of time based on historical prices with 68.3% confidence). Variance reduces to realized volatility rather than implied volatility (the expected percentage move over the next year based on option prices with 68.3% confidence) since it is calculated with measurements—as opposed to being implied through supply/demand imbalances by the options market. 

One interesting fact regarding the dynamics of variance is that, since volatility is proportional to the square root of time, variance is simply proportional to time.

Variance should always be fully respected as a counterparty force because it is wild and theoretically infinite. Deep out of the money long options like to have a stake in how wild those infinite possibilities can be. As Benoit Mandelbrot said, and frequently quoted by one of the most successful quants of all time (William Eckhardt), “variance is infinite.” Eckhardt explains this in the sense of coin tosses, and how drifting 10 in a row in either direction can get the mean significantly off course. From here it can still go any which way, but plausibly drift either way infinitely. 

The importance of that to us as traders is that when we are short options with uncapped risk and no means of hedging surprise moves (such as with stop-limits in place), then we are taking a counterparty to infinity, and that is not a good logical place to be without safeguards.