Canon EOS 5D Mark III
In "A" auto mode everything is going to be done for you.
In "P" program mode you'll need to set some things. It is likely set to a single focus point or a small cluster of focus points, which is what many professionals and enthusiasts use.
Press the af selection button (far right button near the top - looks like the plus sign made of dots) and then quickly press the M-Fn button on the top. If you are looking through the viewfinder you will see that it is toggling between different types of focussing - Including autofocus.
Hope that is helpful.
BTW - When I got my first advanced dslr I took the manual to bed every night for a week to understand how it worked and I still refer to it occasionally.
Canon has some interesting videos explaining how the 5D Mark III focus system works on their website.