Body composition is generally a numeric measure of leanness, or a body fat percentage. People trying to lean out, or develop a six pack, should be more concerned with body composition than scale weight. It's possible, especially when starting an exercise program, to gain muscle mass while losing fat, and even to dramatically improve your body composition without seeing much movement on the scale.
Body composition can be measured in a number of ways. It's important with any measurement tool to measure while holding as many variables constant as possible - time of day, workout schedule, meal schedule, hydration level, etc. It's also important to realize that each tool is more consistent relative to itself than to other methods, so a change in body fat percentage between different measurement tools is much less meaningful than a change within the same tool (although even that can be within the tool's margin of error). With all measurements, it's most important to watch the trend over time, and to think of the measurement as a range, than to be overly focused on a specific data point.
Bod Pod - Relatively inexpensive, uses air displacement and weight to estimate body composition. This method is sensitive to hydration levels, and can be variable in back-to-back tests of the same person.
Underwater weighing "Dunk tank" - relatively inexpensive, can be scary for some as it requires holding your breath underwater for a period of time. This method is sensitive to how much air remains in the lungs, so is highly variable even in back-to-back tests of the same person.
Dxa Scan - expensive xray technology, does subject the body to a low level of radiation (they say it's equivalent to eating a banana, so probably not much to worry about). This is my preferred method because it claims to be the most consistent relative to itself, but is still not "exactly right." This one has the added benefit of measuring body composition by body part, and also bone density.
Skin Fold Calipers - inexpensive, can be done frequently, but is subject to a high level of variance, in part due to human error. Best not to be performed by a loved one (due to the high incidence of ensuing arguments observed by yours truly!)
Bioelectrical impedence - this one can be pretty good - it depends to some degree on the quality of the equipment, and how carefully a person keeps the conditions of the test consistent, but taken seriously and done frequently to assess a trendline versus individual data points this can be a good tool. It is highly sensitive to hydration levels. It should be viewed along the lines of scale weight - sensitive to what you ate yesterday, and may not tell the whole picture, but not so much so that you can't trust it at all.
The Best (and Worst) Ways to Measure Body Fat, Health.com