The simplest possible background is a painted wall, or a sheet or blanket hung against a wall. Generally, such a background should have a matte surface to prevent reflections (especially when using fill-in flash).
Try to avoid color clashes with the subject and their clothing. Hot colors (reds, oranges) advance and are best avoided, while cool colors (greens, blues) recede and help the subject stand out. You can buy brick photo background whenever required.
Also try to avoid tonal clashes such as a pale subject against a very dark background, or a dark subject against a very light backdrop. It is highly likely that your camera meter will not cope well with such extremes.
Don't use backgrounds with horizontal and/or vertical lines. A brick wall, for example, makes a particularly bad background due to the severe horizontal and vertical lines, plus the red color of brick is too warm and clashes with most skin colors.
The overall aim should be to minimize the competition for attention between the subject and the background.
If you are unable to find a suitably plain backdrop (or decide not to use one for other reasons), move any unnecessary clutter from the background. Reflective objects in particular are best removed out of shot (e.g. mirrors, pictures under glass).
If possible, use a large aperture setting to put the background out of focus. This has the added benefit of making the subject stand out sharply. The further away the background is, the darker and the more out of focus it will appear.
When outdoors, don't shoot against hedges as a backdrop. They are usually too dark and tend to leak light producing a speckled effect.