In thermodynamics, entropy increase, as contrasted with entropy decrease, can be interpreted in two ways: one the result of a process in which entropy was added to a body; two the result of an irreversible cyclical process, during which time there are "uncompensated transformations", which are quantified by the value N, the "equivalence value of all uncompensated transformations".

History
The concept of "entropy increase" was formulated in 1854 by Rudolf Clausius, as presented in full in his 1865 textbook The Mechanical Theory of Heat, and is meant to take into account the view that a differential quantity heat, symbol Q, dQ, $\bar{d}Q \,$ , or đQ, depending on notation assignment, or rather "caloric", as it was called in Guyton de Morveau's 1787 parlance (or igneous fluid or matter of heat, as it was called in Antoine Lavoisier's 1777 parlance) (or phlogiston, symbol ϕ (phi), as it was called in Georg Stahl's 1703 parlance) (or terrra pinguis as it was called in Johann Becher’s 1669 parlance), is not a type of fluid-like massive or massless indestructible particle (as was believed at the time of the launch the science of thermodynamics, by Sadi Carnot in 1824) acting to cause repulsion (or the force of repulsion) in side of bodies, according to Boerhaave's law (1720), but rather a state of "movement", so to speak, a phenomenon discovered by Benjamin Thompson and his famous Cannon boring experiment (1798) (who curiously and simultaneously not only disproved Lavoisier's caloric theory, the one utilized by Carnot in scripting thermodynamics, but also married Lavoisier's wife) a combination of light (photons) interacting with the atoms (electrons), as quantified in Clausius' day, as best he could, by the measured phenomenon of the mechanical equivalent of heat and the postulate that the mathematical conception of the exact differential for a cycle integral can be formulated for units of heat, such that the phenomenon that heat can be converted into work and work into heat, both inside of bodies in transformation and as a body as a whole, according to a fixed ratio, now known as the "joule" the unit of energy.

Lambert | Energy dispersal
Laymanized views of what exactly constitute "entropy increase" abound.

A noted mistaken view of what "entropy increase" means is the 2000s work American organic chemist Frank Lambert, author of the various dumbed-down entropy for dummies websites (SecondLaw.com, 2ndLaw.com, Shakespeare2ndLaw.com, EntropySimple.com, EntropySite.com, etc.), centered around a 2002 article he published in the Journal of Chemical Education, in which he argued that portraying entropy as disorder is misleading or confusing and should be abandoned and instead replaced with his model of equating "entropy increase" as the being the "spontaneous dispersal of energy" or equating it to the "spreading out of energy", or something along these lines. [1]

Lambert mass emailed dozens of American college textbook authors, from 2000 to 2010, imploring them to change the term “disorder” to “energy dispersal” in their chemical thermodynamics chapter, which, supposedly about three dozen of them did. Hence, there are about 30 different college chemistry textbooks floating about the book market with Lambert’s fallacious definition of entropy increase as energy dispersal.

This view, however, fly's in the face of German physicist Rudolf Clausius’ original 1854 theorem of the equivalence of transformations definition of entropy increase, which itself is derived or rather based on the 1840s mechanical equivalent of heat and justified by Clausius’ proof of impossibility of perpetual motion of the second kind, the idea that if entropy did not increase it would be possible to couple to heat engines together and make a perpetual motion machine.

Lambert, however, admits that he nearly failed his senior year of college thermodynamics, finding the subject so difficult that he was forced to switch majors from physical chemistry to organic chemistry. This lack of foundational understanding of thermodynamics shows through in his attempt to define “entropy increase” simply as “energy dispersal”, which is a fallacious view, and moreover to pass this deficient definition off to American students is nearly a crime against humanity, akin to John Neumann telling Claude Shannon in circa 1948 to call his telegraph wire coding math logarithm formula by the name entropy, because, in Neumann's own words, "no one really knows what entropy really is, so in a debate you will always have the advantage."

Lambert admits that in formulating this model he never read the original Clausius, but culled from his “energy dispersal” model from an article of William Thomson, and his verbalized thermodynamics models of a universal tendency in nature to the dissipation of mechanical energy, both of which lack in the mathematical rigor and foundation of Clausius, which is rooted in the foundations of mathematics, down to William Hamilton, Joseph Lagrange, Leonard Euler, down to Isaac Newton and Gottfried Leibniz, all of which Lambert says should be tossed by the wayaside all because he doesn’t like the word disorder and thinks that the deeper underlying of entropy might be "too difficult" for college chemistry students to comprehend.

This judging entropy to be "too difficult" view, by no coincidence, resulted in the Erwin Schrodinger's ridiculous 1944 concept of "negative entropy" (or negentropy), which is akin to calling a the variable X by the name "negative X" or negX, which never occurs in mathematics, but has been taken up like candy to laymanized thermodynamicists, the world over, all because he judged that free energy might be too "difficult" a term to explain to his lay audience in his lectures, and followup now-famous book What is Life?, all of which he had to recant in his infamous "Note to Chapter 6", where he explains how he was attacked by his fellow physicists for promoting the nonsensical idea of "negative entropy" as a measure of order.

See also
● Surroundings entropy increase
● Local entropy decrease
● Law of increasing entropy
● Maximum entropy
● MaxEnt school
● Entropy of mixing

References
1. (a) Bindel, Thomas H. (2004). “Teaching Entropy Analysis in the First-Year High School Course and Beyond.” Journal of Chemical Education, November, Vol. 81, No. 11.
(b) Thermodynamic entropy (2010) – Hmolpedia treads.
(c) Entropy (energy dispersion) – Wikipedia.