Master: Contains all versions of all items. This is the database that content authors interact with, and is the default database used by the Sitecore Content Ediitor.
Web: This contains only the current versions of items. This is the database that supports the live website.
Core: This database has two purposes:
It contains definitions for the Sitecore user interface (Content Editor, Page Editor, etc.). This is where you would go to add, for example, a new Right-click option to the Content Editor or a new ribbon button to the Page Editor.
It contains the ASP.NET membership tables that drive authentication and security.