I loaded The Sims 3 on my Mac mini and once I hit the play button, the game won't load further than the photo with a plumbob and a green 3 next to it. Can you show me how to fix it?

There have been many versions of the Mac Mini over the years. Which one do you have -- look under About This Mac on the Apple Menu and report back on model, processor, RAM, and Graphics if you could please.

The lower end ones could not possibly run TS3 for Mac, their processors are too weak and their graphics are incompatible. Some of the higher end might be able to, but it's not likely to be a great simming experience. I would imagine that all but the first couple of released EPs would be out of the question even on those. I would also be concerned about cooling -- not sure Minis have a sufficient fan in that little box.