Have you ever watched a movie where something is happening on a Monday and then the next day is Saturday and everyone is on their weekend? Not because time has passed that you haven't been shown, but because they have literally jumped to Saturday without any events occurring in between, on or off screen.
Have you ever seen a character wearing one thing and then the next minute they are wearing something different? Or more importantly, they are eating something and it magically reappears, whole and untouched upon their plate in the next minute.
Continuity errors really jump out at me in films because you have to wonder with the number of screenings they do why nobody picked up on it. Of course, now that I'm writing regularly and finishing my own work, the answer has become quite apparent.
Too many details.
My brain is racing around trying to keep up with the main plot and sometimes small details slip by me without a second thought. Even on the third or fourth reading I don't notice. Sometimes even friends read the story and don't notice the error. Yet there it is in black and white, just waiting to be discovered and mocked.
I'm a lot nicer in my critique of stories since I've been writing.
The biggest problem I have had recently is in weather and the time of day. I particularly had this problem in my second MS (though I had numerous other problems with this MS, including a psychotic protagonist who refused to work as per the plan). The story takes place over six days and is divided into six parts with approximately seven chapters per part.
It all seems wonderfully structured and organised until you realise that chapter three of part one has a character watching a sunset and then in chapter five there is another sunset. Not the same sunset in a different location. Hours have passed and the sun is definitely setting again.
Part three is equally inept with a storm brewing, that never approaches. Not disappears or goes around. There is a storm brewing, everyone is worried about it and then it literally never gets mentioned again. Gone. Unimportant. Except for the reader who flicks back wondering if they missed something.
I also had a problem with directions. If you walk north from this building you end up at that one. Except when you don't. Except when they head off north and exit the city from the same location, not passing the building they used to arrive at when heading in the same direction.
How to solve all these problems? I would say how to avoid them in the first place, except I know that is not going to happen, so now I'm just going to work on how to identify these problems and fix them.
1. Sometimes stories jump around or are not told in a linear fashion. After you've written the first draft, create a timeline of events and make sure that if something hasn't happened yet, it isn't mentioned.
2. Around the outside of your timeline, you might want to list any external phenomenon that are mentioned. Storms, tides, wars, grazing animals, etc. Anything that might give the reader a moment of confusion if it changes or disappears illogically.
3. Draw maps. I hate maps, I won't look at them in books. If I can't get a feel for the place from the writing, I certainly don't want to try and decipher someone's artistic rendering of it. However, for the sake of organising a place in my head, some sort of visual representation of the main areas (main city, main residences, main rooms) will help you sort out your location and spacing problems. Even in rooms, if the chair is against one wall, it can't suddenly be under the window, etc.
4. As to fixing these problems, don't do what I frequently do and change the story at one point without following the correction through. That just creates more problems in the long run.
Continuity errors - don't think no one will notice.