Does anyone know of an open source module or a good method for handling application errors and e-mailing them to an admin and/or saving to a database?
ELMAH is a great drop-in tool for this. DLL in the bin directory, and some markup to add to the web.config and you're done.
You should look into the application_error method in the global.asax page to catch the errors, and into either log4net and the enterprise library to log those errors in whatever form you choose to any provider you choose -- such as database or e-mail.
log4net can save errors to a database or send emails. We use this at my job (despite the fact it caused Jeff Atwood much stress in the SO beta). Catch the errors in the global.asax page in the Application Error method.
log4net is open source, fairly easy to learn, and can be configured to log to multiple sources (file, email, db, eventlog) very easily.
CALM does this and more. It's not free or open source, but it is inexpensive and comprehensive.
caveat: I am the author of CALM
MS Enterprise Library definitely offers this functionality. It has a little bit of a learning curve, but once it's set up, it is pretty easy to maintain. It also offers several different logging sources, but is not open source.
I have a free provider as well that is available from my website that is a drop in solution for error handling.