Although there are a variety of styles of macro photography, a common approach is to try to isolate the subject from its background. To do this it is necessary to achieve a 'clean' background, free of clutter. This is achieved in one, or more, of the followings ways:
1. By ensuring that there is a good distance between the subject and the background
2. By using a large aperture (low f number) which gives a shallow depth of field
3. By using a greater level of magnification. Once you are working close to or beyond 1x magnification, the background will always tend to be free of detail.
Sometimes the surrounding landscape and foliage will make (1) practically impossible. Also, relatively large subjects such as dragonflies or larger butterflies will not require a significant degree of magnification. This means that on many occasions (2) is the only option. The challenge is then to ensure that the whole insect, or at least the important parts of the insect (e.g. eyes), are in focus. This generally requires the lens to be parallel to the insect which is obviously easier when viewing from the side relatively 'flat' insects such as dragonflies/damselflies or butterflies with their wings closed. If the lens is carefully positioned it is possible to achieve an image of a damselfly or closed-winged butterfly that is pin sharp throughout at an aperture of f4 or f6.3.
So, if you've mastered the ability to achieve clean backgrounds the next important consideration is the colour of the background. Sometimes you will have little choice over this but on other occasions simply changing the direction from which you are photographing can dramatically change the colour of the background. Also, you can seek to photograph subjects in certain locations, knowing that the colour of the surrounding foliage will complement the image.
The image below is a good way to illustrate the above points. Since I wanted a relatively clean background but with some subtle colour variation I chose an aperture of f7.1. This didn't provide a great deal of depth of field, especially for a relatively small insect such as this Blue-Tailed Damselfly, and so I had to very carefully position the lens to ensure that both the eyes and the tip of the abdomen were in focus. I then had a choice of whether to photograph the subject against a lake and some reeds or to switch sides and photograph against a field of buttercups. I chose the latter and the result is the striking yellow background.
PLEASE CLICK TO ENLARGE IMAGE
Having spelled out my approach to clean backgrounds, I will end on a couple of notes of caution. Firstly, it can sometimes be the case that a background is too uniform and clean. Sometimes some subtle tonal variation, such as in the above image, is therefore desirable. Secondly, don't fall into the trap of thinking that all macro images require clean backgrounds. On occasions, it can be desirable to achieve lots of detail in the background in order to show the subject in its natural environment (see here for a post on 'wideangle macro').