What is the meaning of "polarity" as it applies to standard household wiring?
Is it really important as everything still seems to work?

 

Means the hot wire (black) always goes to one side of the outlet, and neutral (white) always goes to the other. Can never remember which is which without looking, but it's the reason outlets have one long and one short plug hole. As far as the electrical theory on importance of same, will have to wait for a real electrician to post.

 

For a technical answer from a licensed electrical expert (I'm not one), you need to post your question over in the Electrical forum.
Basically, in standard 120v residential wiring, the black wire is hot and connected to the breaker in the main electrical panel.
The white wire is neutral and connected to the neutral bar in the panel.
You can get killed by either one, if the circuit is hot.
The bare wire is ground and connected to the ground bar in the panel.
For proper polarity on an outlet, the black (hot) wires should be connected to the brass colored screws, and the white (neutral) wires should be connected to the silver colored screws, and the bare (ground) wires should be connected to the green ground screws (and pigtailed to the outlet boxes, if they're metal).
Yes, it will still work if you reverse the black and white wires, but you are exposing yourself to a possibly fatal electrical shock hazard (reverse polarity) with lamps, appliances, etc...or whatever is plugged into it, and maybe a fire.
The long slots on the receptacle are for the neutral side, and the other is for the hot.
There are a critical purposes for doing it (polarity) right. Electrical fires are the number one reason that homes burn down. Improper wiring is why people can get electrocuted.
Good Luck!
Mike

 

The polarity ensures that the hot conductor is switched by a light or appliance, and not the neutral.
Functionally the circuit will work either way, but it is safer to have the polarization.