The ‘rule of thirds’ is an idea that important features are placed off centre in a photo, as this is more aesthetically pleasing in many cases. Many cameras these days can display a grid in the viewfinder, but if not just try to imagine dividing the area into three both vertically and horizontally. Use these lines or intersecting points to position dominant features in the photo. For instance, in a landscape, instead of positioning the horizon in the middle of the frame, position it two thirds up the photo to allow more foreground or one third up the photo to allow more background or sky.
If you want to focus on a specific object or person, half depress the shutter release button then re-compose the frame before pressing the button all the way down. Alternatively you can usually use the dial on the back of your camera to move the main focus point.
Of course everything is subjective and rules are meant to be broken. I saw a stunning sunset of Tindaya Montana the other day with it dead in the centre of the photograph, albeit with the horizon just below the lower third to allow some fantastic clouds.