Persuasive design is one of the many tools in a designers toolbox. It can be applied to products or services in order to nudge or change a user’s behavior towards a desirable (ideally positive) outcome that aligns with their intentions. Persuasive design is the applied form of various insights derived from established Psychology and/or Social studies. The main aim is to motivate the user into making the right decision, something that goes beyond just establishing usability.
Persuasive design tactics are not about coercing or manipulating your users into wrongful decisions. Your aim should always be to help, through understanding the deeper underlying needs of the user and using the appropriate level of persuasion to help them achieve their goals.
One applicable framework for persuasive design is the Fogg Behavior Model (FBM) developed by B.J. Fogg, from Stanford University. Persuasive design can be found in many domains since it has wide applicability, from e-commerce to lifestyle and health there are countless examples and opportunities.
You can find many examples of persuasive design in the biggest e-commerce sites (amazon, ebags etc.) and maybe an abuse of it in social media sites (facebook, twitter).
One example of lifestyle/health is the app QuitNow, designed to help people who want to stop smoking.