There’s little that I could add that hasn’t already been said, but the two biggest improvements an amateur can make is to get closer to their subject and pay attention to the light. Photography is about capturing light (at its very very basic level), so if you have a shitty lighting situation, don’t expect it to look any better on camera (minus lighting the scene yourself).
Robert Capa said something along the lines of if your pictures aren’t good enough, you’re not close enough. There’s a lot of truth to this statement. Much can be improved my simply moving closer to your subject. I would phrase the rule as “fill the frame.” In other words, eliminate any extraneous detail and fill the frame with your point of interest. This simplifies the composition and focuses the viewer’s eye on the details of the subject that interest you. Now, what is extraneous and what is not? That’s a judgment you have to make, but usually a third of the frame with uninteresting dirt or a plain, boring sky are not necessary elements to the photograph.
Some of the most striking pictures (take, say, Steve McCurry’s Afghan Girl or most of Yousef Karsh’s portraits) are dead simple compositionally, but succeed for several reasons, one of them being the elimination of all unwanted detail. Watch those backgrounds! Again, watch those backgrounds! When you photograph something, it’s not unusual to be so focused on the subject that you neglect to see what’s going on behind your subject. When I go to a location to photograph, the first thing I do is to mentally note where the good, clean backgrounds are and where the problem areas are. Unless I’m doing an environmental portrait that is trying to illustrate the clutter (or I otherwise judge it to give the photograph context), I don’t want trash cans in my backgrounds. I don’t want exit signs. I don’t want light poles sticking out of people’s heads. And so on. Pay attention to your environment. Pay attention to what’s behind your subject and in your frame.
Light is a little more difficult to explain in a message board post. Light is the basis of photography. It is our paint. It creates mood and character in our photographs. Simple guidelines: landscapes are generally best at the “golden hour”–the time just after sunrise and just before sunset. The easiest (but least dramatic, usually) light for portraits is on overcast days. When photographing portraits, pay attention to how the light hits the face. Where are the shadows? Are they covering up the eyes, giving your subject that “raccoon” look? (Normally, a bad thing) Is the shadow crossing the nose and bleeding into one of the eye sockets, making one eye look smaller than the other? (Also, usually considered sloppy, with exceptions.) Or is the light good, but your subject is squinting, because they’re looking straight into the sun? Be aware and constantly thinking critically about your light. It’s as important, if not more, than your composition. One of the other things I note when walking into a photography situation is to take note of where the best light is. Sometimes, this will be in the shade. Sometimes in the sun. It depends on the situation. If it’s noon and sun in full force, often a shady location will be the best bet for a soft light. It it’s sunset, and the light is coming in diffuse through the clouds, lord knows I’m going to take advantage of that and place my subject facing into the light, probably at a 45 degree angle or so.
Let’s dissect a photo. Afghan Girl. Obviously, the most important part of the image, that which gives it the soul, is the subject herself and her eyes. I analyze photos on three main criteria: moment/meaning, composition, lighting. I think they’re almost equally important, with (and this is my photojournalism background showing) more weight going to moment/meaning/news value. So what is Steve doing in this picture? Let’s see compositionally: Absolutely clean background. Just a little information: some green, probably a building/structure of some sort, but nothing to distract our eyes. It’s a sympathetic background, with the green fortuitously echoing the colors in her torn clothing. Plus we have these wonderful, organic complementary red and greens, throughout the photo, even down to the eyes. It’s practically a two-color composition. There’s really no rule of thirds even going on in this frame. The subject’s near eye is dead center in the composition. But you still have a wonderful diagonal leading the eye from Afghan Girl’s eye down to the tear in her clothing. It’s quintessential “fill the frame.”
Let’s look at light in the photo. It’s a large, soft light source (you can tell by the quality of the light/shadows on her face itself, as well as by looking for the highlights in her eyes) from the right/slightly top right of the frame. Since the light is soft (I’m guessing an open doorway or something), there are no harsh shadows across her face. Her eyes are brilliantly illuminated, drawing our interest. Since the light is directional (that is, coming across the frame rather than straight onto the face), we get a sculpted, three-dimensional look (rather than the flat look of a direct flash), and we can get texture, patterns of highlights and shadows across her face, folds of clothing, hair. All that, combined with the mesmerizing glance, and we have a brilliant photograph.
That’s the principles above shown in a classic photograph (there’s more I can detail about the composition, the aperture, the interplay of positive and negative space, etc., but that’s more than needs to be said right now.) So, fill the frame, understand your light(ing), capture a moment.
Simple, no? 
edit: Well, my “little” turned into a “lot,” didn’t it? Sorry for the stream-of-consciousness, but I hope it offers some insight. Note my use of hedge words in my post, as there are no real “rules” to photography. Eliminating all background detail is quite often not the best compositional choice – sometimes that context will make the picture, rather than break it. But, for simplicity’s sake, I offer the above points to think about.
