According to halacha (Jewish law), a father is obligated to circumcise his son on the eighth day after birth (unless some health reason requires postponement). The circumcision is to be done by the father or an agent he appoints; there is no requirement that the person performing the ritual be a rabbi, priest, or any other special category. Today it is customary to hire a mohel to do it, but that is a matter of expertise -- most people today do not know how to do it safely. (We also tend to hire agents to do other specialized things, like writing a torah scroll and slaughtering animals for meat. Halacha does not require hiring a specialist; practicality usually does.)
The torah records one case of a woman performing circumcision (Moshe's wife Tzippora circumcised their son on the way to Egypt in Exodus chapter 4). Under the Greek ban on circumcision c. 215-163 BCE, Antiochus degraded and executed mothers of circumcised babies, which makes me think the mothers might have sometimes done the act themselves. Usually, though, I believe either fathers or agents hired by fathers were the ones to perform circumcision throughout our history.
The Mishna, the part of the talmud compiled around 200 CE, talks about when to perform circumcision, but I haven't found anything there about where. Today the circumcision can be held either in the synagogue (typically right after morning services) or in the home. A celebratory meal is part of the mitzvah, so practicality might govern the choice sometimes.
The one performing the circumcision says a b'racha (blessing) before doing so: "Blessed are you, Hashem our God, King of the universe, who sanctifies us with mitzvot and commands us concerning circumcision". The father also makes a b'racha before the circumcision: "Blessed... and brings him (the son) into the covenant of Avraham our father". In some traditions the father also says Shehechiyanu, a b'racha expressing thanks for reaching this time. (It's not specific to circumcision; this prayer is also said on festivals, when donning new clothes, when eating a type of fruit for the first time in a year, and other occasions.)
See the Halachipedia page on circumcision and its many citations for more on the halacha as we have it today.