Aperture Choice and Depth of Field
The aperture is the opening through which light enters the camera. If you haven't read my post regarding the exposure triangle, I recommend reading that before reading this one.
As I've explained in my previous post, the larger the aperture, the more light hits the sensor and the smaller the aperture, the less light hits the sensor. In order to get proper exposure in your photo, appropriate shutter speed and ISO must be selected as well in accordance with the photographer's aperture choice. In addition to just regulating the amount of light in the image, aperture also controls the depth of field. Depth of field means the area that is in focus in the image. Greater depth of field means, that more of the image is in focus, while shallow depth of field means that less of the image is in focus.
When should you use shallow depth of field and when is greater depth of field more appropriate?
Shallow depth of field is often used in order to emphasize the subject of the photograph, while blurring out people or things that don't contribute to the story. For example, when taking a portrait of a person, you may want to blur out the background in order to focus your viewer's eyes on the subject. You may even want to focus on the subject's eyes and blur out the nose and ears. This type of selective focus requires a wider aperture. The following image, I've shot at F3.2. The background is blurry, but still clear enough to give some context to this portrait of a young Lao village boy.

Greater depth of field is more appropriate when you want the whole image to be sharp, or at least most of the image to be in focus. For example, while shooting a landscape, you may want to maximize the area in focus. This can be achieved by using a narrower aperture. Take this Parisian city photograph, in which everything is in focus, because the city is the subject. This was shot with an aperture of F9.

Apertures are marked by f-stops in photography. Counter-intuitively, larger apertures have lower f-stops, while smaller apertures have higher f-stops. F2.8 for example is very wide aperture, while F16 is very narrow aperture. The reasoning for this is that the f-stop number is actually part of a fraction, yet only the denominator is marked for practical reasons. So F4, for example actually means F 1/4. If you're using a 200mm lens, your aperture is actually 1/4 * 200, or in other words 50mm. So F4 for example is actually F 1/4, which on a 200mm lens would mean 50mm, and F8 is actually F 1/8, which on a 200mm lens would actually be 25mm. As you can see, F4 would actually be the larger aperture.
High end lenses tend to have wider aperture choices, such as 2.8 or lower numbers. I even have a lens which has an F1.2 aperture which is considered extremely wide. It also allows me to achieve very shallow depth of field. However, even if you have a lower end lens or camera, there are still some tricks to help you achieve shallow depth of field. The closer you are to your subject, the more shallow the depth of field will be. In addition, longer focal lengths help achieve shallower depth of field. So even if you're shooting with an F5.6 aperture on a 135mm lens and are getting as close as you can to your subject, you will be able to achieve a relatively shallow depth of field.
Here is an image I've just at F5.6, yet from very close to the model. As you can see, her nose and ears are somewhat out of focus. Although I used a similar focal length for this image and the image of the Lao boy and although I'm using a narrower aperture, this image has shallower depth of field.

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Thanks for reading and keep up the good light!
Yehonathan