A dedicated mobile website sometimes called a web app is different from a responsive site and is often run parallel to a desktop website on a sub-domain like m.mywebsite.com Users re-directed depending device and directed to the site that is right for them. Often this is a better solution for mobile users when compared to a responsive design which tries its best to cater for all screen sizes.
When a responsive layout is not precise enough you need a dedicated mobile web app.
A web app is only one step away from a mobile app that you might download from the app store or iTunes that installs and runs on your phone. The main difference is that a web app exists and runs on a web server and uses native website code like a normal website.
Images, media and content are delivered just to suit the mobile user giving them a better, faster experience and usability is not compromised by having to cater for other devices.
Web apps are normally used where compatibility is critical like in an e-commerce website situation where the extra development cost can be justified by more sales.