When taking photographs, the photographer, whether they may realize it or not, is trying to convey something. It may be a general sense of beauty at the vista before them, or it may be a particular emotion that their subject is feeling. The possibilities are limited only by the human imagination. However, not every photograph captures with clarity the sense the photographer intended. One reason that this happens is because of a failure to adequately appreciate and account for the background.
One thing to remember is that even if you already have your subject picked out, you should take note of the surrounding objects that might also be considered subjects. Oftentimes people will point and shoot a picture of their friend when traveling and not take into account that there are other subjects in the background that might detract from the original subject of the photographer. This is true whether you are taking a picture of a person, an object, or a location.
Choosing a complementary background does not involve only one particular method. It is important to remember that one method may work in a particular instance, but that same method may work against the subject in another instance.
A great example is blurring the background. In shots where the subject is lost in a cluttered background, it is a very good idea to blur the background. This lets the subject stand out from its surroundings and the viewers eyes are drawn towards the subject automatically. This is achieved by using a larger aperture setting. At other times, the background should be in focus. In trying to capture the grandness of a mountain range surround a farm, perhaps, you want to make sure that the entire picture is in focus, so as to achieve your desired effect.