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Chazakah is a presumption, particularly of occupation or ownership. You could say colloquially, "I have a chazakah to that parking space; I park there every day", or "I've been hanging my washing in front of your window for 11 years and you haven't said a word, you can't suddenly tell me to take it down now because it's an eyesore; by now I have a chazakah."
The argument, which everyone understands, is that continuous use or occupation creates an assumption about rights of ownership. It is a similar principle to the saying "possession is nine tenths of the law".
This is reflected in the literal meaning; chazakah means "holding", from the word chazak, strong or solid (as in "chazak chazak, v'nitchazek", the encouragement to go from strength to strength which we declare on finishing reading one of the five books of the Torah.) The one with the chazakah is usually the one holding it. The burden of proof is on the person who challenges the chazakah.
A large and subtle body, halachah deals with the question of how a chazakah is established or challenged. For houses, occupancy of three years creates a chazakah (Baba Batra, chapter 3). This length of time is the duration for which we assume many people hang on to their deeds of ownership. After this period we believe a householder who, when challenged to produce a deed, claims to have lost it.