First of all, I'd really like to thank BlueBellFlora for all the work, time and patience that she has put into researching and putting together many tips and tricks for the Mac users. This is a Mac user's Best Friend. Read this thread before posting any problems and I'm sure you'll find an answer to your question or concern. Caspinwall, itsaprilXD and tamalinn also have contributed to this thread and many guides other as well.
The next thread is from caspinwall and details problems and solutions with using Origin. Yes, Origin does work on the Mac side. Yes, it is your best friend if you lose your disk and have registered your game. If there is a problem with a game not showing up in your Origin account and you have registered it, you will need to contact EA support.
And lastly not to toot my own horn, here is a thread about how to patch your game correctly and safely. I am patched to the current patch 1.67 and run all of the EP's and SP's as well as a lot of Store items and sets. My game is stable and runs fine. I haven't tried to run it on high graphics settings lately, but I'm sure it will run fine.
We have another thread here that may help with the Disk Authentication Errors that several folks have had. It leads to the EA Help Forums.
If you are trying to play Sims 3 on a newer Macbook, here is a list of the new Macs which have the Iris Intel Graphics cards. Please be advised that Sims 3 does NOT recognize this brand of graphics card and you may experience weird graphics or the game may not start at all. There is a workaround since Sims 3 will never be updated again.
So Mac users, join me in thanking these wonderful people who have researched and tested these solutions. I know I really appreciate all their hard work and time to write these guides.
EDIT as of April 2018:
Newer MacBooks have a hard time running this game with all the EPs/SPs. You can maybe run with the Base Game and a couple of EPs installed. Be warned that Seasons and Pets are usually the two that will give you the most trouble due to the intense graphics these EPs offer.
Since the release of El Capitan (Mac OS version), physical disks will no longer be recognized. You will have to play the digital version of the game through Origin. If you are the original owner of the disk, you can register the CD key at Origin for free and your digital copy will appear in your library. No need to repurchase the game.
If you find that the game does not run as you hoped on the Mac OS, try Bootcamping your Mac and running it through the Windows OS.
Second Star to the Right and Straight on 'til Morning.