This is yet another term from that repository of extraordinary expressions, the field of rhetoric. It refers to the trick of repeating the last word of a sentence, line, or clause, at or near the beginning of the next.
An example will make the idea clearer and to give it I call upon that fortune-cookie philosopher, Yoda from Star Wars: “Fear leads to anger. Anger leads to hate. Hate leads to suffering.” Understanding you are? A more sanctified appearance is at the start of the Gospel according to John in the King James Bible: “In the beginning was the Word, and the Word was with God, and the Word was God.”
Anadiplosis derives from Greek diplous, double, from which also come diploid, diploma and diplomat (the last two from the idea of a doubled or folded paper, hence an official document). The prefix ana- is also Greek, meaning back or anew.
Do not confuse this figure of speech with epanadiplosis, in which a sentence begins and ends with the same word. A famous example is in a speech by Malcolm X: “You bleed when the white man says bleed. You bite when the white man says bite, and you bark when the white man says bark.” The extra prefix that appears in epanadiplosis derives from the Greek preposition epi, “upon, in addition”.
Likewise, don’t muddle anadiplosis with the better-known anaphora, in which successive clauses or sentences begin with the same word or words:
Fog everywhere. Fog up the river, where it flows among green aits and meadows; fog down the river, where it rolls deified among the tiers of shipping and the waterside pollutions of a great (and dirty) city.
Bleak House, by Charles Dickens. Ait is another way to spell eyot, island.
Another rhetorical term for a similar trick is antistrophe (which is also known as epiphora or epistrophe — there’s disagreement over terms), which refers to repeating a word at the end of successive clauses or sentences (“government of the people, by the people, for the people, shall not perish from the earth”). Both antistrophe and epistrophe derive from Greek strephein, to turn.