The ritual of kaparot [lit. atonement], which is usually done just prior to Yom Kippur (usually the day before), where one waves a chicken or an [unspecified] amount of money over one's head in a circular motion three times while chanting a text, and then gives the ritually used object to charity, seems difficult to understand. If charity can atone for sin, why does one need to wave it around one's head? Can this ritual be performed any time of year? Where did the idea of chickens come from, and why a chicken more so than any other animal?
Nitei Gavriel Yom Kippur 10:2:4 says that a chicken was selected for the following reasons.
A: It is not too expensive.
B: It should not seem like a Korbon (Makriv Kodoshim BaChutz). And a chicken is not used as a Korbon.
Nitei Gavriel Yom Kippur 10:17 says that it can be done also with live fish and money, and you also wave the money around your head.
Nitei Gavriel Yom Kippur 10:7 says that if it was not done during the Aseres Yemei Teshuva it may be done on Hoshana Raba. It seems from here that it is specific for this time of the year.
Kitzur Shulchan Aruch (131:1) states: