Electricity flows in a “circle”, or more properly a circuit. DC electricity flows one way around that circle. AC electricity goes back and forth in a sine wave, flipping which way the electricity is flowing 50 or 60 times per second (depending on what part of the world you are in).
In DC circuits, which wire is which matters. In AC, since the electricity is flipping back and forth in direction, there is no “positive” wire and no “negative” wire. So the two wires are effectively interchangeable.
It would be “safest” (intentionally put in quotes) to leave both of the AC wires ungrounded, since that way you could touch either wire by itself and not get shocked, even if you were touching something metal at the same time. And in fact, this is done for things like hospital operating rooms and other locations that are defined as “wet” locations by the National Electric Code. These are called “isolated” systems since they are isolated from the Earth. If you are in a hospital (hopefully just visiting or for something minor), look for the orange outlets. Those are isolated outlets. Not to be confused with red outlets, which are outlets that are on the emergency generators and remain powered even when the hospital’s power goes out.
While isolated systems are safer, they aren’t practical for residential electrical service. Isolated systems in hospitals have to be inspected every year and go through a lot of maintenance and testing to make sure that they stay isolated, and this is all inside the controlled environment of a hospital building. If you try to run an isolated system in the real world, mother nature is going to randomly make ground connections for you. Instead of an isolated system, you’ll end up with a randomly grounded system.
So what we do in the real world is we intentionally ground one of the AC wires. This is done by literally connecting the wire to a copper rod that is pounded into the ground. Your “grounded” wire is now called your “neutral” wire, and the ungrounded wire is the “hot” wire. Now it matters which wire is which, just because of this ground connection.
A grounded system like this is not quite as safe as an isolated system. You can touch the neutral wire and something metal like a water pipe and not get shocked. But if you touch the hot wire and a water pipe, you’ll get shocked. If the system was isolated you wouldn’t get shocked, but maintaining an isolated system in every home isn’t practical, so it’s safer to have one wire intentionally grounded instead of having mother nature randomly ground one of the wires.
For a long time, this was how homes were wired. You only had two pronged outlets, with one of the prongs being the “hot” wire and the other being the “neutral” wire (the wider one is the neutral).
So let’s say you have something that has a metal case, like an oven or a refrigerator, or maybe even an old fashioned metal vacuum cleaner. If the “hot” circuit inside the device accidentally touches the metal case, then if you touch the case and something grounded like a water pipe at the same time, you’ll get shocked (and possibly killed). So that’s bad. So they used to connect the neutral to the case. So now it’s safe, right? If the hot wire has a fault and touches the case, all it does is blows the breaker.
But here is the problem. What if your neutral wire breaks? Now if you turn on the device, the case floats up to 120 volts. This electricity is going through the device so it’s not quite like a direct short circuit to the case, but it’s still potentially deadly if you touch the case and something metal like a water pipe. So that’s bad.
To fix this problem, they started using 3 wires instead of 2. Instead of using the neutral wire as both the electricity return wire and the safety ground at the same time, now the neutral wire only carries the electricity back around the “circle”, and the separate safety ground is used to connect to the metal case to prevent you from getting shocked. The safety ground is the little round prong that we’re talking about in this thread. The safety ground is also connected to Earth ground, at the same point where the neutral wire is grounded.
With this 3 wire system, if the hot wire breaks, then the device stops working. The metal case remains safe to touch. If the hot wire shorts to the case, the device blows the breaker, and the case is still safe to touch. If the neutral wire breaks, the device stops working, and the case is still safe to touch (unlike the 2 wire system). If the neutral wire shorts to the case, the case is still safe to touch. If the safety ground breaks, then as long as you don’t have any other fault, the case is still safe to touch. It requires more than one thing to break for the case to become dangerous to touch, unlike the 2 wire system where just a broken neutral connection can make the case unsafe.
If you break the ground connection off of the plug, then you already have your one fault. Now, all it takes is a single other fault and you can end up being unsafe. Yes, “it still works”, because that’s the beauty of the 3 wire system compared to the 2 wire system. But if you do have another fault now, instead of it remaining safe, now you could potentially kill someone.
If you get it fixed, then no matter what breaks, as long as it’s a single fault, the worst that can happen is the device stops working. If you don’t get it fixed, the worst that can happen if something else breaks is that someone dies a horrible painful death.
As was already mentioned upthread, it’s called the “safety ground” for a reason.