In thermodynamics, reversible, as compared to irreversible, refers to a process in which the forces involved in the expansion (forward process) of a body exactly equal the forces involved in the contraction (reverse process).  A reversible process is a hypothetical construct; originally outlined as a realistic process by French physicist Sadi Carnot, in his 1823 Reflections on the Motive Power of Fire, based on the now defunct caloric theory of heat, then later modified in differential form by German physicist Rudolf Clausius, beginning in 1850, using the mechanical theory of heat.

Reversibility
A process is called reversible if it takes place as a succession of “equilibrium states”, i.e. a hypothetical or incremental condition in which the internal expansive force of the body is assumed to nearly equal to (or be larger by a differential amount) than the external pressure force of the atmosphere or surroundings, e.g. a connecting vessel or vacuum; where the reverse process is assumed to leave the surroundings unmodified. A reversible transformation is said to satisfy three criteria: 

(a) The transformation follows a perfectly defined path at all points.
(b) The reverse transformation, following the same path, must be possible at any moment.
(c) If the system is made to undergo the forward process and then the reverse process following the same path, the work exchanged with the surroundings must be equal to zero: The third criterion means that the transformation must occur without any friction; its presence, more or less unavoidable, prevents actual transformations from being reversible. 

Incorrect | Views
In 1997, French biophysicist Louis-Marie Vincent, in interview, in a discussion about near-death experiences, wherein people die then come back to life, incorrectly described the expansion and contraction of an iron bar as being an example of a “reversible phenomenon”, as such: 

“Take the simple example of an iron bar which, when heated, lengthens and dilates. If we let it cool, it will contract and return to its original size. This is what is meant by a reversible phenomenon.”

He then asks us to consider whether or not death is a reversible phenomenon? Vincent’s example of the heating and cooling of an iron, however, is not a reversible process, in that the measure of the entropy change of the body will be positive at the end of the cycle. In fact this is the very same example used by French physicist Sadi Carnot in his 1824 Reflections on the Motive Power of Fire, an example later explained to be irreversible by German physicist Rudolf Clausius.

References
1. Clausius, Rudolf. (1879). The Mechanical Theory of Heat, (pg. 30). London: Macmillan & Co.
2. Perrot, Pierre. (1998). A to Z of Thermodynamics, (pg. 265). Oxford: Oxford University Press.
3. Elsaesser-Valarino, Evelyn. (1997). On the Other Side of Life: Exploring the Phenomenon of Near-death Experience (ch. 6: “Dialogue with Louis-Marie Vincent, PhD”, pg. 174). Plenum Press.