What type of statistics is used to describe this chart? What kind of data analysis technique / formula would you use to describe which is the fastest OS in this chart, Windows or Linux? 

If Linux and Windows had equal speed on all tests, all bars would be 50% blue, 50% green.
The data is duration in milliseconds. Each number (e.g. Linux Startup) represents the mean of 1000 individual duration tests on that scenario. The results of each test individually is covered in separate charts. 
This data represents an overview of the one set of data (Linux over the course of these 5 tests) and how it relates to the second set of data (Windows over the course of these 5 tests). The aim is to be able to say:
Over the 5 tests we Windows is faster. (You can see this in the chart because there is less green, but this is to prove it statistically).
Note:
The chart below shows the same data in real value terms, showing that one test dominates the others. While this is an important result, it does not weigh on the current question as the two results in the 'Refreshing Desktop' test still bear the same relation to each other as those in the other tests.
In the comments there has been some discussion on how to represent this data. The reason I represent it in stacked bars is because I want to show it without the overshadowing effect of the one anomalous data. Is there a preferred method to show this relationship?

 A: Displaying the results
You said in comments 

I used stacking purely to display relationship of data. The actual quantity (duration) is irrelevant 

Stacking makes the length comparison difficult; for the ones that are close together in time, you'd need to look at the numbers to see a difference. If the values don't matter, one option is to plot the logs of the ratios instead (but with axes representing the percentage difference)
With a plot more like this:

you get a better sense of the relative speed of the various tasks under the two systems.
However, it may be worth looking at plots where all of the individual values are represented (rather than just averages), so that an idea of the relative variability is also available.
Testing for a difference
The issue with any kind of statistical test is being very precise about what it is we mean to test.
In particular, what null and alternative hypothesis we're dealing with, and under what assumptions.
You have 5 numbers for each type of activity - are they just repetitions, or is there some kind of pairing across 5 different circumstances? Are you looking for a test within each kind of activity or something overall? 

R code to (approximately) generate the above plot:
wl <- read.table(stdin(),header=TRUE)
Linux Windows Scenario
1.962 1.415 "Startup"
8.469 6.996 "Shutdown"
102.2 79.3 "RefreshDesktop"
1.777 2.866 "AccessInternet"
1.259 1.165 "WebcamStartup"

wl2 <- as.matrix(data.frame(wl[,1:2],row.names=wl[,3]))
wl3 <- wl2[,2]/wl2[,1]
opar <- par()
par(xaxt="n")
dotchart(log(wl3))
abline(v=0,col=8,lwd=2)
axis(side=1, at=log(c(1/1.6,1/1.4,1/1.2,1,1.2,1.4,1.6)),
    labels=c("60%","40%","20%","same","20%","40%","60%"),xaxt="s")
title(main="Relative speed of Windows and Linux",
   sub="             Linux slower by                        Windows slower by")
par(opar)

