1. Aperture Priority / Shutter Priority
Many traditional photographers swear by shooting in Manual mode at all times, but it's extremely helpful to know all of the modes available to you and what they're best used for. Different camera manufacturers have different names for their modes, but the two that are most helpful are Aperture Priority (Av for Canon, A for Nikon) and Shutter Priority (Tv for Canon, S for Nikon).
Aperture Priority allows you to set the aperture you want, and allows the camera to automatically configure all other settings of the camera. This is a great way to control your depth of field by using the largest or smallest aperture available on your lens at any given time.
Shutter Priority, on the other hand, allows you to set a shutter speed and have all other settings of a camera be automatically determined. This is perfect for getting crisp shots of fast-moving action or slowing down the shutter for panning shots.
On older cameras, you'll still need to manually set the ISO. Most newer cameras have an auto ISO feature, so make sure to check your manual.
2. White Balance
Auto white balance is great for most situations, but sometimes the best course of action is to take control and set a white balance that matches your situation. Most cameras have a collection of pre-set white balances to choose from for a variety of shooting environments, but the two presets that are most commonly used are Tungsten and Cloudy.
Tungsten (usually indicated by a lightbulb) balances your photos with cool colors. If you're noticing that a lot of your indoor photos have an unattractive yellow hue, try using the Tungsten setting to create more natural-looking colors.
Cloudy (typically indicated by a cloud) adds a yellow hue to your photographs. If you're shooting outside on a cloudy day and your photos all look blue and dreary, the cloudy setting can create photos with a more balanced color palette.
3. Continuous Focus
Cameras are usually set to Single Focus when they're first taken out of the box, meaning that when you press the shutter halfway the camera will focus on a single point until a photo is taken. However, Continuous Focus (Servo for Canon, AF-C for Nikon) allows the camera to automatically track a subject moving toward the frame. This setting is essential for taking multiple photos of a subject quickly moving towards you.
5. Back Button Focus
Most DSLRs allow you to assign a button on the back of the camera to focus, instead of using the main shutter button on top of the camera. This is great for low light situations where your autofocus may get confused and start searching for a subject every time you take a photo -- just use the back button to focus once, then whenever you press the shutter button a picture is taken.
A slightly more advanced way of using back focus is to completely separate exposure lock from focus lock.
6. Exposure Compensation
Exposure Compensation allows you to lighten or darken the entire frame of the final photograph taken by your camera. Properly using Exposure Compensation can help alleviate a lot of headaches that come with using Aperture or Shutter Priorities.
If you're taking a photo that you know your subject will be dark (for example: a portrait against a bright sky), you can use Exposure Compensation to brighten the entire frame, which will brighten your subject. However, be aware it will make ALL parts of the photo brighter. Be careful not to over expose!
7. Flash EV Compensation
Flash EV Compensation allows you to control the power of your flash to make it brighter or darker. If your flash is too bright and harsh indoors, try using Flash EV Compensation to bring the power down. Alternatively, you can raise the flash's power outdoors to increase the effects of using a fill flash.
8. Camera Beep
Nothing makes your presence as a photographer more obvious than the annoying beeping sound that cameras make when they've locked focus. Luckily, this feature can be turned off in your camera's settings. This is especially important for photojournalists or street photographers. Check your camera's manual to see how to disable the feature on your specific camera.