In a purely scientific sense, entropy deals with energy - more specifically, the lack or "unavailability" of energy in a closed system. Physicist Rudolf Clausius coined the term "entropy" in 1850 to describe the unavailability of heat or energy.
Another manifestation of entropy is the degree of randomness or disorder in the universe: the tendency of systems (and the universe is the largest system of all) to degenerate from order into chaos. Since the second law of thermodynamics says that entropy is continually increasing, the universe is in essence "breaking down," transitioning from a state of order to chaos.
H.G. Wells is describing a stage of entropy at the end of The Time Machine (1895), with the huge red sun. H.J. Campbell wrote House of Entropy (1953).
In a more poetic sense, entropy is used to denote a system or thing's decline or breakdown. Entropy became the dominant metaphor for the British "New Wave" science fiction (as opposed to American SF's metaphor of space travel) featured in Michael Morcock's New Worlds beginning in 1964 and best characterized by Pamela Zoline's "The Heat Death of the Universe" (1967).
Philip K. Dick wrote often of entropy, as in Do Androids Dream of Electric Sheep? (1968), where he calls it "kipple" Entropy is a major theme in such works as Thomas Pynchon's Gravity's Rainbow (1973), Robert Silverberg's "In Entropy's Jaws" (1971), and Michael Moorcock's The Entropy Tango (1981).
In Stephen King's Dark Tower series, the people in Roland the Gunslinger's plane of existence or dimension say that "the world has moved on" - expressing the feeling that things in the universe are slowly falling apart and breaking down.
Entropy as a metaphor for decline and decay is also prevalent in the short stories of J.G. Ballard - such as "The Voices of Time" (1960) - and is expressed in genetics: animals integrate lead into their shells to shield against radioactivity; the last fish alive on Earth struggles to survive in a shallow pond where a leaking dam threatens to spill the remaining water; and humans, unable to cope with the future, become genetically programmed, when genes within their cells become activated, to sleep almost 24 hours a day.
The most famous science fiction work about entropy (or at least with the word entropy in its title) is Geo Alec Effinger's novel What Entropy Means to Me (Doubleday, 1972). In the book, the entropy in the universe is physically concentrated at a specific location, the "Well of Entropy."
Michael Moorcock also used entropy as a major theme in his 1981 novel The Entropy Tango. Isaac Asimov writes about entropy in his 1956 short story "The Last Question," in which a supercomputer figures out a way to revive the universe after its heat-death by entropy.