Keep in mind that jQuery Mobile isn't "jQuery for mobile". It isn't a replacement for the jQuery library for mobile use; rather, it is a user-interface library built on top of jQuery. You must include jQuery, as well as the jQuery Mobile stylesheet, to exploit its capabilities.
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 it works best 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 for desktops, also built on top of jQuery. Learn more at jqueryui.com.