First off, the Business Productivity Online Suite is often called BPOS for short. Much easier to say, but it only helps if you know what it is.
BPOS is an offering by Microsoft as a Hosted Solution- also called Software as a Service (SaaS). Enough acronyms, we don't want to scare you off.
BPOS is a suite, as the name implies, of several different products:
While each can be purchased ala carte- Microsoft has seductively priced the collective as a suite.
As a Hosted Service, everything is managed and stored online by MIcrosoft. Email, for instance, can either be accessed via a web browser or using Outlook if preferred. Documents (ie: Word docs and Excel spreadsheets) can be stored in a SharePoint share. It's still required you have desktop licenses of MIcrosoft Office to view/edit the files, but their organization, management, and access control is done through SharePoint Online. Similar are the services of Office Communcations Online and Office Live Meeting.