Complex. Have you read the book? The explanation in there is much better than the one I'm about to give.
Ketosis is a fat burning state.
Normally, your body uses sugar for fuel. Not just yummy white sugar and sweeties and things that taste sweet: all carbohydrate is treated as sugar by the body. So if you're eating pasta, you're eating sugar. A baguette? Sugar. Baked potato? Sugar.
The body loves all these things because it's quick easy fuel for it to use - and back when food was hard to find that was a great thing. We ate an apple, the body processed it into sugar and we kept functioning. Whee!
The problem comes when we have a ready supply of carbs - sugar - so our body is easily getting all the food we need. We're in a sugar burning state and any excess food we eat is packed away neatly on our bodies for use in the hard times that may come later. In other words, we get fat. It's like coming home with a bag of coal and some logs for the fire - the coal (sugar) burns easiest so the logs get stored away for later (as fat).
But when those hard times do come, and there's no more coal, the body has to start using up its fat supplies, the ones it stored away like logs.
When you're starving, you go into ketosis. It's a state which allows the body to use fat as fuel. You know what tells your body it's starving and is allowed to do that?
No more sugar.
This means that, on Atkins, as long as we cut out sugar our body will go into that fat burning state of ketosis and burn off our stored fat supplies. It will be conditioned to use fat as fuel. This state takes a few days to get into and as much as a few weeks for the body to get really really efficient at.
That's the problem with cheating on Atkins. As soon as you eat carbs (sugar) or even something that isn't carby but triggers an insulin response that makes the body -think- its getting sugar, your body thinks the starvation period is over, comes out of ketosis and starts burning sugar as fuel again. It refills the body's glycogen supplies and if you eat carbs for long enough, starts refilling its fat supplies too. Then it will take you a few days to get back in and start losing fat again.
Ketosis: the state in which the body burns fat for fuel instead of sugar.
(it's more complex than that but unless we're diabetics it doesn't really pertain to us - this is just the explanation as I personally understand it).