The most confusing part for beginners is that aperture is an inverse value. The lower the number, the larger the opening.
So as you can see above, an aperture of 2.8 is larger than an aperture of 16.
The size of the aperture opening determines Depth of Field or the amount of a scene that is in focus. Bokeh is an effect of Depth of Field and changes according to the lens diaphragm size and shape. There are also filters one can make to shape it as well.
The larger the opening, the less that is in focus. The easiest way to remember this relationship is the larger the number, the more that is in focus.
But Wait! There's More! The focal length of a lens will also effect depth of field. A wide angle lens will have a larger depth of field than a telephoto lens at the same F/stop.
Here's an image captured at 300mm(450mm equivalent) at F/5.6.
Here's another still at F/5.6 but at 18mm (27mm equivalent)
Yet another factor in depth of field is the size of the medium or sensor size. Point and Shoot cameras have a sensor roughly 15 times smaller than a DSLR. So even at telephoto ranges the depth of field is pretty long. An APS-C sensor is large enough for depth of field to be fairly obvious.
Common Sensor Sizes
Depth of field is important when composing your image. With a shallow depth of field, you can make the subject stand out and prevent any distracting details in the background disappear. Or if you want to have a landscape with all of the features sharp and in focus, use a larger depth of field.
The third factor is camera to subject distance. The closer your focus is, the shallower your depth of field.
Take some time to experiment, take multiple shots of the same scene with different depths of field and see which one appeals to you.
Aperture also is the main factor in controlling flash intensity but that's for another time.
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