What's the best way to clean my bathroom tiles?

Here’s a pic of a clean bathroom tile:
http://s8.postimg.org/kf1vyy1x1/DSCN9387.jpg

Now here’s a dirty one:
http://s8.postimg.org/naez5t5x1/DSCN9386.jpg

Here’s a clean and dirty one side-by-side:
http://s8.postimg.org/4fj8fe5v9/DSCN9382.jpg

And finally, here’s what I think is bathroom mold:
http://s8.postimg.org/qdfp96kvp/DSCN9384.jpg

I want to learn to become a typical handyman, and I really want to clean the dirty tiles. I don’t just want to scrub off the mold or whatever’s on there, but I also want to make it look as white as the unaffected areas and also a way to prevent this from happening – if not permanently then at least from time to time.

I tried the cleaning products I have at home with little result. I need something stronger or more recent.