The Coolest Button
A DSLR's shutter button is super super cool. It's basically the coolest button. It, unlike most buttons, has not one, but two (!) degrees of "click". It's awesome!
Pushing the button halfway down will engage a camera's auto-focus mechanisms. You'll probably hear the lens spin a little bit, then - when the camera decides it is in focus, it will beep at you. Most DSLR's will flash little points in your viewfinder when it reaches focus, telling you "these points are in focus".
Try it now. Push the button halfway down and listen for the beep.
I often turn off the beep on my cameras, because it's kind of annoying. If you don't hear anything, check your settings. Either way, look for the viewfinder dots to blink.
Once the camera beeps, it won't change the focus (or the exposure settings) until you let go. You can move the camera around and take the photo, with the focus still at the distance of whatever was in focus when it beeped. This is important for the "focus and recompose" shooting method. Right now, relax, let your camera beep at you all it wants.
Once the camera has finished beeping at you, you can push the button down the rest of the way, and it will take the photo.
With A DSLR, the mirror inside the camera is bouncing light up to your eye, but that needs to move in order for you to take your photo. The mirror swings out of the way, the shutter opens and closes, and the mirror flops back down. The viewfinder goes black for this time, since the mirror flipped up.
That's why cameras blink black when taking pictures. Neat.
Manual Focus Mode
Sometimes the camera just can't seem to focus on what you want it to. Most cameras assume you want to take photos of the closest thing, for example, so it can be tricky to autofocus on something that isn't in the foreground. One of the simpliest ways to get around this is to shoot in manual focus mode.
There is a switch on your lens to switch between auto (AF or A) or manual (MF or M) focus mode.
Switch that to M and find the focus ring on the lens. It's usually the ring furthest forward on the camera. Spinning it one way focus on things further form you, and spinning the other way focuses closer. The specific directions to spin are different depending on the lens manufacturer (Canon and Nikon are opposite, of course).