A continuity error is the side effect of the time dialation effects that video editing cause.
Imagine that you are filming a couple having dinner at a restaurant. You do the scene in multiple takes. The first take is done from over the man's shoulder, looking at the woman, for a 1-shot. The second take is done in the opposite way. Why two separate shots? Because otherwise, one camera would be in the other's shot! A third take is often done in a wide 2-shot, showing both people, and the table.
Okay, now there is a glass of Merlot in front of the woman. It is a really good wine, and saying all those lines makes her thirsty, so she takes a drink during the second take.
Okay, so now the first take shows the glass full, and the second and third takes shows the glass a little lower.
Now, if you interleave the three takes together to get that ping pong effect usually seen when two characters talk to each other, you will see the wine glass appear to empty and fill itself.
It is subtle, but some people (see continuity fascist) will actually watch for these things. This is also only one example. You will also see things like blood splotches moving, wet clothing becoming suddenly dry, then wet again. Broken glass fixes itself, etc, etc.