On-post is a little ways from post but is surrounded by a fence and you have to show ID to get in, just like post. There are a few (very few) officer houses on Villagio.
Here is a VIDEO of an on-post apartment.
Government leased are places that the government rents and puts people in, the government pays the utilities and deals with the landlord. Typically if there are any government leased places available they will force you to take one, you cannot choose private housing unless all the government stuff is full because they don't want them sitting empty when they are already paying for them. There are different houses available through government lease, some that are meant for enlisted, some upper enlisted and some officer. Enlisted are typically in an apartment building with a shared yard whereas upper enlisted and officers are sometimes in a duplex or something that has it's own yard.
Private housing you rent yourself from a local landlord, housing helps you work out a contract but you are on your own for utilities. If you are in private you get an allowance for utilities but it can be difficult to figure out sometimes as they don't consistently send your utility bills and sometimes add random charges. But it is nice to be able to have more of a choice in where you live when looking for housing. Check out www.ahrn.com to see the listings of private housing. They have listing for all the different bases in Europe so if you want to get an idea of what housing is like where you will be moving, check it out! If you are wondering how much your overseas housing allowance is check here: http://www.defensetravel.d
When I talked to housing they told me that they are in the process of approving a plan that will force ALL officers out of Villagio and government lease (not force them out now but not allow them any new contracts...) Officers will have to get a private rental. That way there will be more government housing available for enlisted. This is not in place yet, but is in the works...as far as I understand :)