This question is a bit misleading - when querying, inserting, updating, or deleting information from a database through an ASP page, ASP is not what is, directly, working with the database. Rather, ASP is utilizing a set of COM objects referred to as Active Data Objects, or ADO for short. These objects were designed to connect to a wide array of data sources. Furthermore, since these objects are COM objects, they can be accessed from any programming technology that can utilize COM: ASP, Visual Basic, Visual C++, Visual J++, the Windows Scripting Host, etc.
So a more correct question is what databases can ADO work with? The noon-technical answer: a lot!! The more technical answer: any data source that provides an OLE-DB or ODBC interface can be accessed by ADO.
OLE-DB is an application programming interface (API) for accessing data stores; OLE-DB serves as middleware, sitting between an application that needs to interface with a data store and the actual data store, facilitating communication between the app and the data store. A data store, as its name suggests, is a store of data. A relational database, like SQL Server or Access, is a data store - an email message or Microsoft Index Server are both data stores too, though. For more information on OLE-DB be sure to check out OLEDB.com and Microsoft's OLE-DB site.
ODBC (Open DataBase Connectivity) is an older data access standard developed by Microsoft. ODBC was Microsoft's attempt at creating one standard database protocol - when using ODBC an application could issue the same commands whether the database was SQL Server, Microsoft Access, or Oracle. ODBC is an older technology that has been replaced by OLE-DB. However, you will often see databases being referred to as ODBC-compliant. You can access these types of databases (ODBC-compliant databases) through an ASP page using ADO.
So if ODBC and OLE-DB are two middleware products, where does ADO fit in? Well, it too is a middleware component, another level of encapsulation for developers to utilize when needing data access. ADO presents data access in an object-based paradigm. ADO sits on top of the OLE-DB, handing off data requests to OLE-DB.
So, back to our original question: what databases can one use on an ASP page? Again, our technical answer is: any data source that provides an OLE-DB or ODBC interface. So what data stores provide such access? Tons of 'em. Here is a quick list with links to learn more about how to work with these data sources:
To connect to some of these data stores you need to have the proper ODBC or OLE-DB drivers installed on your computer. These should be freely available from the database provider's Web site. (For example, the Oracle ODBC drivers can be downloaded at: http://otn.oracle.com/software/utilities/software_index.htm.)