No version of Lightroom runs in the cloud. They all run as a local application on your computer.
No version of Lightroom stores your photos in the cloud. They all require you store your photos on a local drive or on a network drive.
Adobe sells two versions of Lightroom, one called Lightroom 6 (which is 79 US dollars as an upgrade or 149 US dollars as a first time purchase) and another version called Lightroom Creative Cloud 2015, which is 9.99 US Dollars a month and has more features than Lightroom 6, includes Photoshop, and allows you to share your photos with mobile devices (tablets and cell phones).