Problem

While saving I get the error message: An unexpected error has occurred while saving: Error 12 (or 13 or 15 or 16). Please consult the readme for details.

Meaning of the Errors

Error 12: World file save failure. Probably Memory related
Error 13: Post-save callback failure. Unclear what that means. They fixed an error that was causing this and it seems to not occur as often.
Error 15: Open failure. You probably don’t have enough rights to write in the folder.
Error 16: Loose file copy failure. The file cannot be (over)written.

Avoidance & limitation of possible loss

Since Error 12 seems to be caused by memory issues, a 64-bit operating system and at least 4 GB of RAM help avoide this error. EA altered the executable file with Patch 1.17/etc so that the game can allocate 4 GB of RAM.
If you are on a 32 bit system you can try the suggestion here Game Help:Insufficient Memory Crashes, Step 3. I do not really recommend it, especially for non-tech people but you can do so at your own risk.
Make sure your Computer fulfills the minimum requirements (EAs minimum requirements are a little low, especially with all EPs)
Save often and in cycles (5-10 savegames with different names or use Twallan’s Saver Mod to auto save.
Regularly backup your user files.
Check the folder CurrentGame.sims3 before starting the game. It should be empty. It only contains data during gameplay. If your game crashed, this folder might still contain something. Move or delete it’s content.
Make sure your saves folder doesn’t contain any saves with the extension .bad. Delete those, they are unusable.
Increase your virtual memory
Start your game in compatibility mode and/or as administrator
Make sure you have enough room on your hard drive
Defrag your System
Don’t play long game sessions. Quit and restart the game in between.
If the error occurs regularly in one game but not so much in others, try cleaning your game with Twallan’s Error Trap or move your sims to a new town with the porter (the mod lets you keep your relationships to all Sims you pack to move to the new town)
CC and mods can cause all kinds of Error. Use the 50/50 Method and available tools like Dashboard or Custard to find the culprit
The memory system introduced with Patch 1.21/1.22 bloats your save game enormously. This can become heavy on your RAM. Delete memories for your active/inactive Sims that you don’t need. MasterController helps with that
Since Patch 1.42 you can turn off automatic memories.

Suggestions

You can try the following suggestions while the game is running and then try saving again.

As soon as you get the error, the save is written to a folder with the extension .bad. You’ll find it in your saves folder. Delete that folder before continuing.

Error 12 & Error 13

Use save as and a new name instead of save
Try saving in Edit Town mode as well as live mode
Try saving your game paused and unpaused and at different speed levels.
Move your saves folder to the desktop
Never use save & quit. Always save your game first and then quit the game without saving.
Don’t use spaces or special chars in your save game names.
If you have lots of stuff in your inventory, reduce it.
If all fails, quit your game without saving and restore a backup.

Error 15

Error 15 occurs when you don’t have enough rights to access the saves folder. Make sure you have full access to the folder your saves folder is in and it’s not hidden. If nothing helps, start your game as an administrator.

Error 16

Error 16 occurs because of a file system failure which can occur if you try to overwrite a file in use or that is corrupt. Saving as with a new name should solve the issue. Also make sure you have enough disc space available.

Save Errors (Error 12, 13, or 16)