Aqeeqah and khitan are frequently discussed as though they were one event, and in practice families sometimes combine them, but they are distinct in law and in what they are for.
Aqeeqah is a thanksgiving. On the seventh day after a birth an animal is slaughtered, the infant's head is shaved, silver equal to the weight of the hair is given in charity, and the child is named. The hadith collections report two animals for a boy and one for a girl, and the schools of law differ on the weight of the whole thing: the Hanbali position treats it as close to obligatory, the Hanafi as a commendable custom, with the others between. The meat is divided between the household, the neighbours and the poor, which turns a family celebration into a distribution and is the feature most often emphasised in preaching about it. A version of the rite existed in pre-Islamic Arabia; what was dropped was the smearing of the sacrificial blood on the child's head.
Khitan, male circumcision, has none of that fixity. There is no appointed day. There is no consensus that it is obligatory rather than a strongly encouraged element of the natural disposition described in a well-known hadith. It is performed on the seventh day in some Arab families, at seven years in much of South Asia, and anywhere between in Turkey and Southeast Asia, where it is commonly made a substantial public festival with the boy dressed and paraded. Converts are not required to undergo it, a point of settled agreement that surprises people.
Female genital cutting is a separate matter again and does not belong in the same sentence as either. It is not commanded in the Qur'an, is rejected as un-Islamic by many senior scholars and by the religious authorities of several Muslim-majority states, is practised in a minority of regions with a range of cultural histories that are not all Muslim, and is opposed by the World Health Organization as a violation of the rights of girls and women. Conflating it with male circumcision, in either direction, obscures both.