The Bioelectrical Impedance method of measuring body composition isn't extremely inaccurate and can have too many variables that can cause inaccurate fluctuations. Previous posts are correct about electronic "pulses" being sent through you body, and calculated based on what the pulse is "going through" or around. (Example: bones, muscles, fat, water, etc..) One of the many problems with this can be the effects of potassium/sodium imbalances in muscles tissue based on diet or training, water retention or dehydration, etc. However a pro-argument of bioelectrical impedance is that it is more accurate that WHR (waist /hip ratio) or the really, really old Metropolitan Height / Weight Chart. You remember, the one that says if you're 6"00" you should weigh __ this much. That was created in 1943 to determine longevity for Life Insurance
Skin calipers can be fairly accurate but again as stated above these are only as good as the individuals performing the test. I've tested people with 3 site, 5 site and 7 site measurements. Obviously, a 7 site measurement, all other variables being the same, is going to be the most accurate. But this really does take practice. Lots and lots of practice. So ask your trainer or whoever is performing the test, how many times they have performed this test, and where they trained?
In my opinion, the most accurate way to determine body composition is either by Hydrostatic Weighing, or Computed Axial Tomography. Unfortunately, both methods can be very expensive. A CAT scan is a fairly common machine in hospitals and doctor's offices, however good luck trying to find a technician to perform this to determine your body fat. Hydrostatic weighing is another excellent method, which I've found to be consistently accurate time and time again, however there is the issue with finding a lab, university, etc. who will perform this.
If you can, call around to some of the Exercise Physiology departments at your local universities. Surprisingly, graduate and sometimes undergraduate students need to fulfill course requirements and are required to perform these tests a certain number of times to show proficiency. Another option might be to find sport specific labs or sometimes elite performance rehab centers have these machines. Or lastly (off the top of my head), if you live anywhere near Denver or Colorado Springs, the Olympic Training Facility I know has everything needed to perform both tests. Sometimes they have interns who need the testing experience for a Physiology, or Kinesiology degree.
Sorry that turned out to be such a long reply, I just kept going!