The tradition of Chanukah gelt (money giving to children during Chanukah) originates from a 17th century practice of Polish Jewry to give money to their small children for distribution to their teachers. Later, children were allowed to keep the money for themselves.
In the 18th century, it became custom for poor yeshiva* students to visit homes of Jewish benefactors dispensing Chanukah money. It is also possible that the custom evolved from Jews in Eastern Europe giving coins to religious teachers as a token of gratitude. (Similar to the custom of tipping service people on Christmas.)
In 1958, the Bank of Israel issued commemorative coins for use as Chanukah gelt. That year, the coin bore the image of the menorah that appeared on Maccabean coins 2,000 years earlier.
Children often use chocolate gelt to play dreidel* with. Parents, grandparents or other relatives give older children actual money.
In Chassidic communities, the rabbis continue the practice of distributing small coins to those visiting them during Chanukah. Chassidic Jews consider this to be an auspicious blessing from the Rebbe, and a segulah* for success.