A digital platform is any web based platform for presenting content(things like Facebook, Twitter, Blogs, Websites, and sometimes SMS. This is in contrast to an analog platform (like billboards, direct mail, telemarketing, events, word-of-mouth, etc)
A campaign is the messaging that you would use a platform (or many) to communicate. This could promote a product or service and could communicate it over both digital AND analog platforms.
A brand is your overall messaging that transcends every campaign. It is your reputation or story. You identity.