Further following up the source given by Wikipedia actually answers the second part of your question as well.
The source (Earliest Known Uses of Some of the Words of Mathematics (S)) says
The Arabic translators in the ninth century translated the Greek
rhetos (rational) by the Arabic muntaq (made to speak) and the Greek
alogos (irrational) by the Arabic asamm (deaf, dumb).
This was translated as surdus ("deaf" or "mute") in Latin.
It has more, but the interesting fact here is that the Greek for "irrational" got literally translated into Arabic for "dumb" and then literally into Latin as surd, which again is used for irrational numbers! (This reminds me of the story of the word sine, originating in Sanskrit jiva, turning into Arabic jiba, being written as jb, being read by Latin translators as the Arabic word jaib meaning bay, and being translated into Latin sinus for bay.)
It goes on to answer the second part of your question:
According to Smith (vol. 2, page 252), there has never been a general
agreement on what constitutes a surd. It is admitted that a number
like sqrt 2 is a surd, but there have been prominent writers who have
not included sqrt 6, since it is equal to sqrt 2 X sqrt 3. Smith also
called the word surd "unnecessary and ill-defined" in his Teaching of
Elementary Mathematics (1900).
G. Chrystal in Algebra, 2nd ed. (1889) says that "...a surd number is
the incommensurable root of a commensurable number," and says that
sqrt e is not a surd, nor is sqrt (1 + sqrt 2).
So there's no clear definition. This is clear from looking at various other sources:
(arithmetic) An irrational number, especially one expressed using the √ symbol.
Wolfram MathWorld (emphasis mine):
An archaic term for an irrational number.
There's even a Language Log post called "Ab surd" about this and other meanings of surd.
I think we'd all be better off if the word stopped being used altogether, or at least was always used with an accompanying precise definition.