I'm not a professional photographer in any sense of the word. In the literal sense, I've never been paid to take photographs. My only "training" in photography came in the form of a black and white course in high school where I mostly took blurry close-ups of flowers and my boyfriend, and enjoyed a false sense of superiority as I bathed photosensitive paper in various chemicals (usually for too long).
Light is king in photography. Harnessing its power is one of the most important things you can do to take better photos.
Avoid flash at all costs. Unless you're a pro with a studio and multiple light points, it will generally just flatten your photos. Instead, seek out natural light in all its forms. This requires patience and experimentation.
Typically, you want your light source behind you or off to the side. When taking photos indoors, it's generally best to stand near a window and point the camera into the room.
You can also manipulate limited light sources, like fire, to highlight something specific in your photos.
CompositionIf you're just getting started, the rule of thirds is a good way to look at the composition of your photos. Using this rule, each photo can be divided into nine sections. The greatest point of interest is along the intersection of two lines separating each section. (If you're interested, you can read in more detail here.)
Experiment with taking photos of the same object from different angles. Use negative (empty) space strategically to focus in on something specific. Go in close, back away, climb up on things, crouch down to get a variety of perspectives and train your eye. You can also use cropping after the fact to improve the composition of a photo.
Variety in SubjectThere have been countless occasions where I think something (building, landscape, sunset) is "so pretty" I start snapping photos like crazy. Yet later, when I look through them, they all look boring. The way to draw attention in a photograph is to give the eye something specific to focus on. I like to put something human into each of my photographs: a whole person, a hand, an item of clothing or a man-made object of significance. It's one way to give your photos dimension and draw the eye in, rather than have it rest aimlessly on a flat (albeit pretty) landscape time after time.
The truth is, you'll likely have to take many pictures to find those few gems. As long as you're not dealing with film, this typically isn't an issue. If you have a DSLR, it can be helpful to set your camera to Continuous Mode so that you can take multiple images in close succession. This is especially helpful when taking photographs of people, whose faces are constantly changing and shifting.
As an additional note, all of the photos in this post were taken with your basic point-and-shoot digital camera during my family's trip to Europe in 2010. So although a bunch of fancy equipment can certainly help (if you actually know how to use it) it really isn't necessary to taking good photographs. Besides, who want to live in constant fear of pick-pocketing while lugging around expensive and heavy gear on their travels?