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During the concluding pasuk of each book of the Torah it is customary for the congregation to stand as the Torah reader reads the final words in a dramatic manner, he signals to the congregation, who then respond with "Chazak, chazak, v'nitchazeik" (Strong, strong, and let us be strengthened). The phrase of exclamation appears to be based on Shmuel Bet 10:12, which is chazak V’NITCHAZAK. To say V’NITCHA - ZEIK is  a more correct pronunciation. It would seem the reason that the 'EIK' ending is used in what is called the pausal form, is that the word is at the end of a sentence or otherwise 'in pause' (at an 'etnachta' the equivalent of a semi colon).

The reader then repeats that phrase, it is considered a special honour to receive this Book completing Aliya.  Some say that the person who receives the Aliya should NOT say the phrase, as this would constitute an interruption between the Torah reading and his concluding bracha or, possibly, that it is to him that the congregation says Chazak.

Rabbi Hayim Halevy Donin writes regarding the significance of this proclamation: "This is a cry of encouragement to continue with the reading of the next book, and to return to this one again in due course. The triple use of the word "Chazak" may  symbolize past, present, and future.1

It may be derived from the similar and stronger adjuration of the angel Micha'el to Daniel (Daniel 10,19). Here Micha'el says "chazak vachazok, uchedabero imi hischazakti" (be strong, yea, be strong, and when he had spoken unto me I was strengthened).2

The Rema writes that this custom is derived from the words of Joshua to the children of Israel, "chazak ve'ematz" (be strong and courageous) -- Joshua 1:6,7,9,18.3

Some have a custom to say the word "Chazak" three times since the numerical equivalent (gematria) of the thrice repeated chazak is, numerically, 345;  which is equivalent to  "Moshe."

The inspiration for saying these words in connection with finishing a section of the Torah may have come from the Talmudic dictum that "Four elements need to be strengthened in man: Torah, good deeds, prayer, and occupation." (Talmud - Brachot 32b)" 

Another interpretation of the meaning is: Let us gather courage to live in accordance with the teachings contained in each of the five books of the Torah 4

As for origin: this is more difficult, the custom would seem to have originated in Provence since the earliest reference to it is in the 'Hamanhig' of Avraham haYarhi of Lunel who lived at the beginning of the 13th century.5
This was a period of persecutions and book burning.  A period when great courage was required to read Torah in public, a period when to continue to study Torah may have resulted in the loss of the readers life.

(1) http://www.aish.com
(2) Ezra Tepper, http://shamash.org/listarchives/mail-jewish/volume31/v31n35
(3) http://www.ottmall.com/mj_ht_arch/v31/mj_v31i40.html#CJM
(4) http://headcoverings-by-devorah.com/OrachChayim/HebglossKh.html
(5) Rabbi Silverman.