I played like 3 levels of Dungeon Keeper and was able to save in the first save slot. But after the 5th or 6th level, I couldn't save during gameplay. I get an error message, something like "system error", when I try to save mid-game.

EDIT: I first noticed the issue the level before Snugglebell, I believe it's called Lushmeadow.

During the first few levels there seemed to be at least 2 levels I could choose to play on the level select screen. Now I only get to choose the next campaign level.

This couldn't possibly have anything to do with the fact that I used the creature transfer power up that lets you take one creature from one dungeon to the next level, could it?

No, that shouldn't have any effect.

Is the save button greyed out? Do you click the save button icon and nothing happens? Are the save slots not clickable or something? Have you used all 8 slots?

Once you fill all the slots you have to over-write one to save again. Should just mean click on the save slot bar and hit Enter (can type a new savename if you want).

The game displays an error message when I try to save. I've only managed to save in the first save slot once. All the other save lots are unused.

Thanks DeMignon,I tried your solution and sure enough it was windows "protecting" me from fun games.

Wonder why it let me save the first time though? Maybe running the game from the installer has different privileges?

Yes, indeed. As the installer must be run with Admin rights, starting the game from there gives you the same privileges.