<<Disclaimer:  Verify this information before applying it to your situation.>>
 
Not medical advice
There are two classes of antibodies seen in untreated celiac disease.
Antibodies directed against a fragment of gluten called gliadin
and antibodies directed against some tissue in the body itself,
endomysial(the covering of muscle), reticulin ( the framework for kidney
and liver) and there are some other.
The actual tests are done using blood from the patient. < 5ccs of serum
is usually plenty.  The blood cells are removed.
The gliadin tests is usually an automated machine read test.  While this
means there is little room for interpretor error, there are no standardised
tests, normal ranges, or even methods in use in the US.
The endomysial tests are more dependent on the experience and ability of
a pathologist in looking at a pattern of staining produced by the
patients serum on a slice of monkey esophagus.  While this test is done
in similar way in most labs there are differences in how these are
interpreted.
How good are these tests?
If all of the tests are positive then they are pretty accurate, <90%
right.  However there are several reasons and circumstances when they are
not so accurate.  IGA and IGG are two diffrent varieties of antibodies we
have in our immune systems.  The IGA gliadin and IGA endomysial tests are
the most accurate and also become negative relatively quickly after
stopping gluten ( 3-6 months).  The IGG is not as specific( it can
positive in non celiacs).  However it is important to do both, as about 4%
of celiacs have no IgA at all.
The biopsy is still considered the goal standard to confirm the
diagnosis. \
If one is screening for celiac disease it is important to make sure that
the patient have been on a gluten containing diet.
The tests may also be useful in following up a known celiac to help
confirm that the diet is free of large amounts of gluten.
Also blood tests results may not be directly comparable from one lab to
the next.
Needless to say the interpretation of mixed results some positive and
some negative is complicated.  also the interpretation and use of these
tests in infants may be different.
Not medical advice
Joe Murray