jQuery Mobile is "is a HTML5-based user interface system designed to make responsive websites and apps that are accessible on all smartphone, tablet and desktop devices." With thoroughly tested code that works consistently across all major phone platforms, support for touch and other gestures, and an Ajax-based page transition model, jQuery Mobile offers an easy way to build highly mobile-optimized sites and Web applications.
jQuery Mobile makes building mobile websites and mobile Web applications quicker and easier, with WYSIWYG builders, code tested on lots of devices, and easy-to-add UI widgets. But one need not, of course, use jQuery Mobile to use jQuery in mobile development, and one need not use jQuery at all to develop sites optimized for mobile viewing.
Does jQuery Mobile work for all devices - desktop, tablet, and mobile? jQuery Mobile is certainly more mobile centric than most other frameworks; you'll have to decide if jQuery's strategy - "A unified user interface system that works seamlessly across all popular mobile device platforms, built on the rock-solid jQuery and jQuery UI foundation" - works for your site or application on desktops as well. Sites built with jQuery Mobile will certainly work on desktops; the library has been tested on all major desktop browsers.
jQuery UI is a similar user-interface library, built on top of jQuery, aimed at desktops. Learn more at jqueryui.com.