What is intuitively a syndrome in error correction?
The word syndrome was introduced into error-control coding theory 
and practice (by Hagelbarger) very early on. It is a term borrowed from
medicine: a syndrome is a group of symptoms, e.g. stomachache, nausea,
fever, lack of appetite, which together are diagnostic (or highly suggestive)
of the presence of a specific disease or condition -- in this instance, appendicitis.  Note that the presence of a single symptom is usually not
considered diagnostic of a specific disease, it is the presence of the
syndrome or a collection of symptoms.
This notion in medicine is almost exactly mirrored in
the error-control coding world. If the syndrome is identically
zero (there are no symptoms whatsoever), nothing needs to be
done, and the received word is accepted as correct (no disease
is diagnosed and no treatment is prescribed).
A nonzero syndrome indicates that the received word has errors
in it (the presence of symptoms means that something is not right
with the patient), and in most cases, the syndrome is also diagnostic
of the locations of the errors and what needs to be done to correct them
with the (Reed-Solomon) decoder substituting for the physician's
thought processes (from syndrome, find the error locations and values)
and actions such as prescribing medicines or performing surgery
(correct the errors, fill in the erasures, etc.).  In some
cases, the fact that the syndrome is nonzero indicates the presence
of errors but the decoder is not able  to determine the error 
locations and values (and so is unable to correct the errors).
This is called decoder failure.  In even fewer cases, the syndrome
indicates a specific error pattern as the most likely cause but in fact the
actual error pattern is something else. Correcting what the decoder
thinks is the error pattern introduces more errors into the
received word, thus making it "wronger" than before, but
this is not detectable at the receiver. This is not called decoder
malpractice but decoder error or undetected error, and is,
fortunately, a very rare event, much rarer than decoder failure.