Is acceleration an absolute quantity? I would like to know if acceleration is an absolute quantity, and if so why?
 A: Absolutely not. An observer in free fall and an observer in zero gravity both experience and observe no acceleration in their frame of relevance. One, however, is actually in an accelerating frame of reference.
A: I've finally figured it out. 
First, let's define precisely what it means for some quantity to be absolute or relative. In the context in question, it has to do with whether a quantity is absolute (that is, has the same value) or relative (that is, has different values) when measured by two inertial observers moving with respect to one another.
Of course, first we need to define what an inertial observer is: it's an observer for which Newton's laws are applicable without having to resort to adding fictitious forces.
Ok, so now we have two observers, Alice and Bob, both of which are inertial. They both observe the motion of some object. Let the index 1 correspond to quantities measured in A's reference frame and the index 2 correspond to quantities measured in B's reference frame. The position of the object is clearly a relative concept, since
r₂ = r₁ + u t
(where u is the velocity of Bob with respect to Alice, and is constant since they're both inertial observers). Note that the time, t, is the same for both observers, as it must be according to Newtonian Mechanics. The object position is a relative concept because r₂ ≠ r₁.
Now, take the time-derivative of both sides and we get
v₂ = v₁ + u
that is, the velocity of the object with respect to one observer is different than the velocity of the same object with respect to the other observer. Hence, velocity is a relative quantity in Newtonian Mechanics.
Next, take the time-derivative of both sides once again, and we obtain
a₂ = a₁
(since u is constant). Thus, the acceleration of the object is the same in both reference frames. Acceleration, therefore, is absolute in Newtonian Mechanics.
When we take into account the theory of relativity, then time flows at different rates for different inertial observers and the result above for the acceleration is no longer true. 
A: In standard Newtonian mechanics, acceleration is indeed considered to be an absolute quantity, in that it is not determined relative to any inertial frame of reference (constant velocity). This fact follows directly from the principle that forces are the same everywhere, independent of observer.
Of course, if you're doing classical mechanics in an accelerating reference frame, then you introduce a fictitious force, and accelerations are not absolute with respect to an "inertial frame" or other accelerating reference frames -- though this is less often considered, perhaps.
Note also that the same statement applies to Einstein's Special Relativity. (I don't really understand enough General Relativity to comment, but I suspect it says no, and instead considers other more fundamental things, such as space-time geodesics.)
A: Acceleration will be the same in any two frames that are moving with constant speed with respect to each other (and may also be rotated and translated).
However, if you consider two frames that have relative rotation or acceleration,  the acceleration of an object will be different in the two frames.
A: Its not, and its explicitly spelled out in Einstein's principle of equivalence  

`We arrive at a very satisfactory interpretation of this law of
  experience, if we assume that the systems K and K' are physically
  exactly equivalent, that is, if we assume that we may just as well
  regard the system K as being in a space free from gravitational
  fields, if we then regard K as uniformly accelerated. This assumption
  of exact physical equivalence makes it impossible for us to speak of
  the absolute acceleration of the system of reference, just as the
  usual theory of relativity forbids us to talk of the absolute velocity
  of a system; and it makes the equal falling of all bodies in a
  gravitational field seem a matter of course.
— Einstein, 1911`

