You don't open iCloud. It's not that kind of service. Instead it is integrated with your apps and other services to sync, backup, and enhance them in several ways. If you were thinking of it as another files storage site like DropBox or Box.com, you're going to disappointed. It might add similar services someday, but not now.
You can access some of iCloud's data in a computer browser at icloud.com, but this only consists a a few apps that sync with your contacts, calendar, Mail, and iWorks apps. Oh, and Find my iPad/iPhone.
Apple - iCloud - Your content. On all your devices.
You can turn the various services on/off in Settings > iCloud, and/or in the settings for apps that use iCloud.