What exactly does being upside down on your loan mean?

Exactly like your question states, you owe more than the house is worth.

Yes, being upside down on your loan describes your current unfortunate situation. This is relevant and usually problematic because if you were to sell your home, you would likely only get enough from the buyer to cover the current value of the property which sounds like would not be enough to pay off your current mortgage. And as it is required to pay that mortgage off to pass good title, that means you would have to pay the difference at closing. It is also still a concern because even if you don't try to sell it for the above reasons, the Lender was using your home as collateral for the loan in case you default. If their collateral no longer protects their investment, default would not be favorable to them either. As to what you can do, there are options directly with the Lender such as a loan modification, special forbearance depending on your situation (e.g. unemployed), short sale, or a deed in lieu.

The lender has total discretion. If you want to keep the house then continue to pay the contract payments.

That is the definition of "upside down," also called "underwater." The house is worth less that the principal balance of the loan. I don't know what you mean by government loans, but you should be talking to your lender about a loan modification. Unfortunately, a lot of people get the run-around in that process too. But to save your house, it's worth a try. Call your lender about it. There are federal laws that govern lenders' processing of these applications.