Ad Extensions are add-ons to Google Ads. Google is constantly adding and changing the options for extensions. Currently the types of extensions that are availabel for ads vary depending on what type of ad you are running. For search ads you have the option of adding call outs (Free Delivery, Licensed and Insured, etc), call extensions (where people can click to call), and location extensions that help your ad show in the map pack. The Display network and YouTube carry their extensions.
The purpose of extensions is to help customers notice your ad and take action.