Autofocus: (AF)
This form of focusing is most commonly found in point & shoot cameras and some more modern SLRs. The camera, or the lens, will automatically focus on your chosen subject when you press the shutter button.
Focus free or fixed focus:
All disposable cameras and some point & shoot cameras use this focusing method. The focus cannot be adjusted but is instead set to a medium distance, keeping everything from a few meters to infinity in focus.
Focus free lens on an Olympus Pen EF Half Frame camera
Zone focusing:
Most famously used by the Olympus Trip 35, plus some other point & shoot cameras. Before pressing the shutter you will need to choose between 3 or 4 different zones. This allows you to quickly set the most appropriate distance, ensuring your photo is sharp and in focus.
On an Olympus Trip 35 the zones are:
- Headshot
- Photo of 2 people from the waist up
- Group photo
- Infinity
Each zone equates to a different distance which is often shown on the underside of the lens
Split image:
Found in almost all SLRs this focusing system is more accurate than the others that have been explained so far. When looking through the viewfinder you will see a ring that splits your image in half. If your photo is out of focus, these 2 images will not line up. Rotate the lens until these 2 images line up and your image should now be in focus.
Rangefinders:
Rangefinders are one of the quickest and most accurate ways of focusing your image. When looking through the view finder you should see 2 identical images. Line these 2 images up by rotating the lens or by moving a lever found just below the lens. Once these two images are lined up then your photo will be in focus. 
Explanation of how to focus a range finder camera taken from a Leica M6 user manual