Web parts are ASP.NET controls that can be added to web part zones. They are generally distributed through the deployment of .WSP files. Pre-2013, they could be developed off the hosting server. With 2013, Web Parts for the most part have to be developed on the hosting server for proper library inclusion.
Web parts let authorized end users modify the content, appearance, and behavior of web pages directly from a browser.
You can read more about how to add, remove webparts in sharepoint, here.
Questions having the web-part regards design, development, configuration, implementation and use of web parts.