It's hard to talk about tools without talking about when to use them and how. With some tools it's easiest to understand them by first understanding why you'd need to use them.
The Clone Stamp Tool is one of the most interesting tools for fixing photos. It is especially good for restoring old photos or poor scans. The clone stamp allows you to paint in one part of an image with pixels "cloned" from another part of the same or a different image.
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Healing Brush tool
The healing brush is very similar to the clone stamp, but it is specially designed for faces or other more subtle tasks for the clone stamp. When you "stamp" with it, it subsequently blends what you've done to make it match the surrounding pixels. Again, it mainly designed for subtle and small areas, like faces.
Also, the healing brush has slightly different brush options than most of the brushes in photoshop. You can easily (and you'll want to) change the hardness of the brush. I recommend a soft one.