Saving problems

Are you supposed to play Knytt at once? When I close the game and I open the game another time to play it, I have to start all over. This is really annoying. Is it because of some glitch or...? 

Do you actually save on savepoints?

There are these white items on the floor that allows you to save (play the tutorial again, and look for the written text to explain everything. You have to press down on them to activate.

If you do that, and it still is not working, then you probably are running in somekind of write protection on the knytt game.

Also note, if you die, you will be brought back to your last savegame. If you collected an item, but not saved, and then died, you will loose the item too.

Another common problem causing is that you might've not actually extracted the game from the zip file.
As Windows treats zip files as folders by default (XP and higher, I believe), you may've not noticed it.

I believe Knytt won't save under a limited user account. Try running it under administrator.

I don't think the user account type should matter, as long as it's in a folder where the user has writing rights.
Of course, if the game is in a folder where the user is not allowed to write in, then it naturally wouldn't be able to save the progress.