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Probably the most central issue that is pertinent to this story is the ebbing and recovery of human civilization, particularly western civilizations. The concept of the Foundation would not exist if it weren't for the efforts of a character determined to ensure that humanity would rise again from the ashes of a collapsed empire. The two stories presented within Foundation and Empire are during the collapsing Galactic Empire's dark age, where entire quadrants are in open rebellion to create their own despotic and barbarous kingdoms. Then, the book answers the question of reunification with an answer that nobody truly enjoys except a few - dictatorship once again. I believe that Asimov attempted to communicate that reunification of humanity with haste will lead to great suffering, even if the promises were worth fighting for.

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Near the very beginning of the second half the story, a clown named Magnifico stumbles upon two of the main characters, Toran and Bayta. At first, his nature ranges from frightened to hysterical, full of despair from being on the run from the Mule. The clown speaks in an odd sentence structure and is a talented musician, and is quite amusing to those who meet him. His antics stem from being the Mule's personal toy for amusement for so long, along with being misunderstood by every person he knows. However, by the story's close, his personality becomes something different entirely - to say how exactly would be to compromise the ending and how it plays out.

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To call the ending of Foundation and Empire a plot twist is an injustice. While very subtle hints were sparsely placed throughout the book, it was not really possible to piece them together on a first-reading. The ending reveals a great deal about mysterious organizations and characters, but renders the entire story pointless if revealed due to the compromising nature of that knowledge. However, the ending was very solidly written and opened up possibilities for a seamless transition to the sequel (Asimov's Second Foundation).

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One of science fiction's primary tropes is the colonization of other planets and the expansion of mankind throughout space. The Foundation series is no exception to that idea - humanity has colonized so much that Earth is a forgotten planet, and not mentioned at all in this entry. In reality, humanity is unable to colonize other planets due to lack of nearby planets and lack of technology to terraform them. However, scientists still maintain hope in looking for earth-like planets. Indeed, scientists have recently discovered a solar system of earth-like planets with relative proximity. A news article is included:
http://www.popularmechanics.com/space/a25336/seven-earth-like-planets-trappist-1/

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While not quite as perfect nor near as tested as Seldon's psychohistorical equations, Cliodynamics is a real attempt to predict the future using data and trends from the past. The creator, Peter Turchin, says that it works if people treat history like a science. More information can be found on his website:
http://peterturchin.com/cliodynamics/

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When Asimov created the concept of the Galactic Empire in his earlier works (Pebble in the Sky, Foundation, etc.), he stated that he based it on the Roman Empire's rise and fall. Indeed, the sacking of the capital planet is similar to that of the sacking of Rome in real life, along with their subsequent decrease in power. A history of the Roman Empire is found here:
http://www.ancient.eu/Roman_Empire/

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The underlying force behind the Foundation's rise to power and the Empire's decline is the mathematical field of psychohistory, developed by Hari Seldon during the events of the first novel, Foundation. Psychohistory is the ability to use statistics and other equations to determine the fate of a society and people, and has served the Foundation well until a certain event during the novel.
Image source: https://s-media-cache-ak0.pinimg.com/originals/c5/9a/d2/c59ad2bd4ad2fbacd04017debc679ddb.gif

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At its height, the Galactic Empire was said to rule the entire Milky Way galaxy, from end to end. However,  by the beginning of Foundation and Empire, it has weakened to its central core worlds, and is essentially a collection of 20 worlds by the beginning of the second half.
Image source: http://www.universetoday.com/wp-content/uploads/2013/10/milky_way.jpg

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The Mule is the pseudonym of the second half of the book's antagonist. During the book, he is often spoken as a gigantic man with the strength of a mule, stylized on this front cover.
Image source: https://upload.wikimedia.org/wikipedia/en/1/19/Foundation_and_empire_cover.jpg