>> I know I'm showing my ignorance here
I think those are very reasonable questions for this forum.
If your just starting Java I would not worry about Database specific knowledge. Actually JDBC is relatively simple once you have a good grasp of java and have even minimal knowledge of how SQL and databases work.
As far as requirements for a specific job, sure that could happen, but I would think even mild experience such as yours with databases would prove enough given the right level of Java knowledge.
As far as what Java is used for besides Applets in web pages. Applets in web pages are probably one of the least used areas in Java. Most of the momentum is in developing server side code. Middleware in Application Servers for things like web development (Servlets, JSP), legacy application integration, workflow, messaging, e-commerce etc. There is also some activity in the mobile/wireless application space but that is probably minor in comparison.
I have not been looking much recently but I think you will find lots of mention of J2EE experience in job postings. Sometimes even a specific Application Server is requested. Since J2EE is a standard one should not need a lot of specific product experience but the App Server products tend to, ummm… do their own thing, yeah same complaint about Microsoft… ssshhhhh don’t say that to loud, oh boy now I’m in trouble! LOL
One thing to keep in mind is that XML is fairly pervasive in server side development these days so experience with XML and the XML DOM and Parsers in Java might also prove useful.
I don’t know if that helps but there it is.
Good luck
-pete