Electrical outlets gone bad?
Welcome to the Magic 8-ball:

Harper calls these the Magic 8-Ball because they sometimes produce meaningless, seemingly random, results. But for straightforward wire swaps they are exactly the tool to use.
These testers make what you found out the hard way a bit easier - i.e., you can plug them into each 120V receptacle to very quickly find out what is going on. What you have would be described by one of these testers as Hot/Ground reverse.
The normal setup is:

A (wide slot) = Neutral
B (narrow slot) = Hot
C (hole) = Ground

You normally have:

A-B = Neutral to Hot = 120V
A-C = Neutral to Ground = 0V
B-C = Hot to Ground = 120V

There are a lot of different things that can go wrong. The most common things are simply wires connected in the wrong places. If Hot and Ground are reversed then you will get:

A-B = 0V (Neutral to Ground)
A-C = 120V (Neutral to Hot)
B-C = 120V (Ground to Hot).

With this specific combination, 2-prong devices will not work but at least will be safe (no voltage, so no current) and 3-prong devices will not work and will not be safe.
To fix:

Turn off breaker
Check color of wires going to breaker, neutral, ground (they will all be &quot;together&quot; either in one cable (Black = hot/White = neutral/bare = ground) or in conduit (color = hot, white/grey = neutral, bare or green = ground or no ground if metal conduit). If the colors don't seem right here (white to the breaker or any color other than white to the neutral or anything other than green or bare to the ground bar) then stop and upload pictures before continuing.
If the breaker seems OK, next step is to open up each receptacle on this circuit and check the wiring. In each case, black or other color (not white or green) should be to the brass screws, white to the silver screws and green or bare to the ground screw.

You will likely find at least one receptacle wired incorrectly. Fix it and then test with a multimeter, though in this case using a Magic 8-Ball tester will speed up the process as this is right on target for that type of tester.