Is the Arctic sea ice disappearing in a linear or non-linear way?
The box in the lower left of the bottom chart is key to understanding what is presented. Five attempts of a line of best fit have been presented. From the chart, the linear fit (yellow line) is the least best fit, so the loss of Arctic ice not a linear relationship. The other lines are different modeling scenarios the author has attempted. 

When testing for lines of best fit, one is trying to find a mathematical equation that will neatly describe something - in this case Arctic ice loss over time.

The other four lines of best fit that were considered were: the exponential curve, Gompertz fit, logarithmic fit and a second order polynomial fit. Visual examination of these curves shows they are very close in describing the data and are close to each other.

The important thing to consider in such curve fits is the value of R2, sometime denoted as r2. A value of 1 means a perfect fit and a value of zero means absolutely not fit.

All four curved lines have a fit greater or equal to 0.902. That is a good result.

The Gompertz fit has an R2 of 0.917 which is the highest value of the options presented. It means that for the data analyzed and evaluated in this study the Gompertz fit is the one that best describes the data. However, given the closeness of R2 values, the other options also give good results, but not as good as the Gompertz fit.