To answer this question it is important to define broad words such as 'success' or 'performance' within the context of the specific application. Key performance indicator is itself a very broad term, with defined sets of KPIs specific to each business and application.
You have to be really clear and careful about what you want to measure then use tools like analytics, tracking, data mining etc. to measure those things. By think carefully I mean qualify the value of a certain measurement outcome, for example is an average short visit time good or bad? Users might actually be very successful at finding what they want, so is a long time on the site really that good?
With e-commerce you will likely be looking at the bottom line: is a satisfactory return on promotion investment being made and the site making a profit? For most applications repeat use is a good indicator of success but most measures will be specific to the case in point.
Talk to your stake-holders and engage your early adopters and this will help you focus on what to measure.