Thought I would add what I've learned myself:
Depth of Field
The DOF in a picture refers to how far objects before and behind your subject are in focus. A shallow depth of field means that only things very close to your subject are also in focus and a deep DOF means that almost everything in the frame - from foreground to background is well focused. Both have their advantages and selecting which one works best with your scene can drastically improve the photograph you take.
By using a shallow depth of field, you can insure that nearly everything other than your subject will appear slightly out of focus. People do not like to look at blurred images, so this forces their eyes straight to your intended subject and holds them there. What a great and powerful tool. What control!
A deep DOF can be used to give clarity to an entire photograph, allowing for multiple objects to remain in focus. Imagine you are taking a photograph of a group of running zebras and you want to see clearly those in the front and those behind. For this you would use a deep DOF. If you only want to highlight the zebra in front and leave the others visually trailing, you would choose a shallow DOF. Two ways to control DOF are by adjusting focal length (how much you have zoomed) and aperture.
All zebras in focus (F/10.0)
This can be a frightening concept for the beginning photographer. It involves the mechanics of the lens and can include mathematical formulas. I won't get into those here, and most of us will never need to. The important basics are that aperture refers to how much the lens opens and is measured in something called F/stops and higher numbered F/stops deliver a greater depth of field. In other words, if you select Aperture Priority mode on your camera and adjust the setting to F16, your photo will have a greater DOF (more of the photo will be in focus) than if you had chosen F4. It's as simple as that!
So imagine you are taking a photo of your favorite animal. While you want to include the other animals around it (if any) in the photograph, you want it to be the unmistakable star of the picture. What to do? Do you want a shallow DOF or deep? Shallow. Set your camera to Aperture Priority Mode, choose one of the F settings with a small number (for example 4 or 5.6), focus on your subject, compose the shot and take the picture. Or if it feels better to you, choose Portrait Mode on your camera, let the camera choose the F/stop, focus on your subject, compose and take your picture. Either way will result in a shallow depth of field, meaning the particular animal will be in crisp focus and the other animals behind it will be visible but less crisp. This will draw the viewer's eye without mistake to your favourite animal. Shallow DOF can also evoke imagination.
Blue Wildebeest in focus and giraffs in background out of focus (F/6.3)
These two techniques can also be used to eliminate distractions from your photographs. Now imagine you want to take a photograph of a beautiful bird in a tree, but the background is full of distracting colors, shapes or lines. You don't want those things stealing attention from your subject, so you create a shallow depth of field to throw them out of focus and get a much better photograph.
Yellow-billed Hornbill (F/4.9)
Now imagine you are in the mountains and want to bring home a photo to remember the beautiful and vast vistas around you. Do you want to create a Shallow or Deep DOF? Deep. You can choose Aperture Priority Mode and set the F/stop to one with a higher number (say 16 or 22), find a point in your landscape to focus on, compose the shot and take the picture.
Misty morning at H10 in Kruger Park (F/8.0)
Alternatively you may choose to set your camera to Landscape Mode so the camera will choose the F/stop for you, find and focus on a point, compose and shoot the picture. Both techniques will give you a photograph with a deep DOF. Something to be mindful of is that when using an aperture with a larger number (or landscape mode) your shutter will remain open for longer, increasing the likelihood of camera-shake. For the best result in this case, use a tripod. If this is not an option, stand with your feet shoulder width apart, hold the camera with both hands and tuck your elbows as close as possible to your body while pressing the camera to your face (use your forehead). You will essentially create a tripod with your body. For even more support, find something (like a tree) to brace your shoulder against while doing this.
For Aperture, Remember;
F/stops with lower numbers = less of your photo in focus (shallow DOF)
F/stops with higher numbers = more of your photo in focus (Deep DOF)
Portrait Mode = F/stops with lower numbers, shallow DOF
Landscape Mode = F/stops with higher numbers, deep DOF
Longer Focal Lengths (more zoom) = shallow DOF
Hope this helps!