There is a custom, called 'Kapporos', which is practiced by slaughtering a chicken for every male, and a hen for every female - and many of the great authorities do not rest easy with its practice. The Ramban and Rashba annuled this custom outright because of its superstitious and pagan-like (Heb 'Darkei Ha'Amory') nature, as can be seen in the responsa of the Rashba (495, and see the Beis Yosef as well). Therefore, our teacher the Beis Yosef writes that one should refrain from this practice.
That said, there are also a great many authorities that endorse this custom, and it is even found in the responsa of the Gaonim. Rashi brings their resonsa in Gemara Shabbos (81.2), where he reports that their custom was to use plants for 'Kapporos'. On the day before Rosh Hashana they would circle plants over their heads and say: 'This is my substitute.' (Heb. 'Zeh Chalifasi') (see there). The Tur, however, reports that their custom was with chicken and rams, and it took place on the day before Yom Kippur.
This is the quote of our teacher the Rema:
"There are those among the Gaonim that mention this custom. Also, one can find many later authorities who make mention of it, and we find it practiced throughout our communities. Therefore, one should not change from following it, for it is a custom of old. The current practice is to take a chicken for a male, and a hen for a female. One takes two chickens for a pregnant woman, for perhaps she will bear a male. White chickens should be chosen, as the verse says: 'If your sins are like an old stain, they will be whitened like snow.' These chickens are then distributed to the poor, or their value in money is distributed to the poor. There are locales where they have the custom to visit cemeteries, and to increase in charity, and all of these are fine practices. One should ensure that the slaughter of these chickens happens very soon after their use in 'Kapporos', and one should place his hands on them as one does a sacrifice [in Temple times]. One should dispose of their innards on the roofs or in the courtyards, meaning in a place where the birds will be able to take them away."
Until here is the quote.
What in particular did some of the great authorities object to? They objected to the preference for white chickens, which is reminiscent of the practice of pagan worship, whose adherents would also search after white chickens for their rituals, as is seen in the Talmud in chapter 'Lifnei Aydayhen' (14.1). Therefore one should not specifically seek after white chickens, though if one happens upon one he can take it (Magen Avraham, comment 3, and the Bach).
One should not take turtle doves or young doves that would be fitting for a sacrifice (in Temple times), so that one does not give the appearance of offering a sacrifice (see there). In addition, one should not follow the instruction to place the hands on the chicken, for that also gives the appearance of being the sacrificial act of 'Smicha'. Even though chickens can not be offered as a sacrifice this is no worse than saying over meat "this meat for Pesach", which is also forbidden, and so how should we permit this outright (Heb. 'Lechatchila)? Therefore, one should refrain from this (Taz, comment 3).
The innards are thrown to the birds as a show of mercy on the creations, symbolizing that Heaven should likewise show compassion on us. It is written that it is a good practice to arrange for the slaughter of the birds in the early morning, which is a time of mercy.
The correct language to recite during Kapparos is 'This is my exchange, this is my double, this my atonement' - which as an acronym is ChaSaCh, in reference to the verse 'who apportions (Heb. 'ChoSeaCh') life to all the living.' The idea is that if death - G-d forbid - had been decreed, this should 'be my double'.
(To my mind it is more appropriate to say 'This is in place of me' instead of the word 'double' (Heb. 'Temurah'), for a "temurah' declaration usually means that both retain their sanctity. (See Gemara Temurah, 26.2.))
The above is regarding the ritual in principle. However, due to our many shortcomings we see now that many of the birds are slaughtered incorrectly due to the large volume of 'Kapporos' and the ensuing pressure. Additionally, the 'Shochtim' (slaughterers) are not able to properly monitor the process due to exhaustion and difficulty, causing many non kosher birds to pass through their hands. They also are not able to inspect the knives properly, and their hands become heavy to them as well.
Due to this, nowadays it is a Mitzvah and an obligation to limit this rather than enter the holy day with all this suspect slaughter. There has already been effort expended in previous generations to annul this practice, though it was not successful because the general populace holds this custom dear, similar to their dearness for the Mitzvah of the Esrog and perhaps even more so. At the very least it should be seen to that the slaughter starts a few days early, and to refrain from slaughter in the early morning, when the hands of the 'Shochet' are heavy.