How can I determine why all the outlets in one room are out?
Something is seriously messed up! If you are not experienced with electrical work do not try to fix this yourself, call someone!

If you are knowledgeable (I guess you must have some knowledge to test like that), then most likely a neutral wire fell off of one plug and managed to land on the hot.

How the hell it did that I have no idea.

Or maybe there is some strange fault in the plug.

The outlets are usually chained together. Sometimes you can get lucky and guess the order of the chain from the layout in the room and where the wires from the breaker panel come in.

Figure out which is the first outlet and look for the fault there.

If necessary pick a random outlet and remove it from the wall and disconnect the wires and cover them with wire nuts secured with a bit of tape so they don't fall off. Then turn the power back on and check which outlets are working and which are not.

Continue in the direction of the working outlets and try to figure out which is the first in line, then test that one.

Good luck, and be safe! ALWAYS test the wire before touching it using a neon tester EVEN if the power is off. (Normally power off is fine, but with a strange situation like this don't take any chances.)