Generally the weights are the most controversial aspects of a decision. Weights specify the relative importance of the criteria. Even after a consensus is reached the natural question remains "what if that one changes", a common way for approval processes to get derailed.
You can change weights by hand to see what happens, but the Weight Sensitivity report will do it for you systematically. It adjusts the weight of each criterion over a range and computes the new ranking, presenting the results in little graphs called sparks like the ones above. The current weight is labeled "At" in the spark. It shows whether changing the current weight a little will change the ranking, or if a big weight change is required. This is not the only way to look at the sensitivity but it does provide a quick way to isolate the items worth discussing.