What is the difference between Sitecore\'s core, master, and web databases?
Some great answers here but it's worth a clear note that the Web database is the default publishing target and is not set in stone. You can rename it to whatever you wish as well as have multiple publishing targets similar to 'Web'.
e.g. We have a Sitecore instance with two websites (website and intranet), thusly the Master database contains all the items for both sites. For publishing we have two targets 'Website' and 'Intranet' that are hosted by different SQL servers.
Ultimately our connections include: Core; Master; Website; Intranet (the latter two being copies of the original 'Web' database of course)
I recommend you look at the Launch Sitecore site to understand more about the CMS.
From the page on Sitecore Architecture:
Master
The Master database is the authoring database - it contains all versions of any content or assets.
Core
The Core database is all Sitecore settings, as well as the tables containing the .Net membership provider (i.e. users/roles contained in the Sitecore repository)
Web
The Web database is only the latest published version, and the content that is driving the live web site. Therefore it is a subset of the master database, optimised for size and speed. When content is published, or goes through the publishing task of a workflow, the latest content version is copied from the master to web database.
Of course, in an Enterprise deployment, you may have multiple Sitecore instances, which may share DBs (or not)