The cloud is the Internet, when used for chores that are traditionally handled by local hardware and software. For instance, if you back up your files to an external hard drive, that’s local. But if you use an online service such as Mozy or Carbonite, you’re using cloud-based backup.
Another example: If you use the installed Outlook program to read email, you’re using the Internet, but not the cloud. But if you read your email on the Outlook.com webpage, you’re reading it in the cloud.