From the consumer of computing services perspective, it means having those services accessible from the internet:
- Online application (e.g. Google Mail, Google Docs, etc.) ;
- Online storage (e.g. S3Fox + Amazon S3, Flickr, Apple MobileMe, etc.) ;
- Online services (e.g. Google Maps, Airline reservation, etc.).
From an IT person or software developer perspective, it means that you deploy your application on a (potential virtual) infrastructure or platform on the internet:
- Virtual infrastructure or grid (e.g. Sun Grid, Amazon EC2, though both have their own specificities) ;
- Platform (e.g. Google Apps Engine) ;
- Identity verification, storage, payments, etc.
Learn more: Cloud computing on Wikipedia