Okay so since Iâve been in a course and a lecture about Roman Religion/Religion in Rome this semester, I might be able to help you.
Since I donât know whether they want to know about a specific ritual, Iâll just start with the general rite for offerings as I know it. (I study in Germany, so I canât guarantee that Iâm translating all the words correctly.)
There are various kinds of offerings, the most widely known being the sacrifice of an animal of any kind, but also the offering of wine or portions of meals. An offering always has to be made by at least three people (tres faciunt collegium), but of course there can be more.
Offerings can be made for the wellbeing of the people of Rome (or any city, I guess) - then they are payed for by the state - or for the wellbeing of a smaller group, like a family, who has to pay for it themselves. The first one would be part of the sacra publica, the latter part of the sacra privata. This determines who is present: normally, a priest (probably from the college of pontiffs) or a commander would be the leader of a procedure that is part of the sacra publica, while for a sacrum privatum the pater familias, the head of the family, would be supervisor.
Everyday offerings were made to the lares, for example portions of meals, at the lararium.
The offerings made on feriae (basically public holidays) were mostly animal sacrifices, though there could be different ones depending on the deity. The most important thing for an animal sacrifice was that the animal itself was 100% flawless. Sometimes there were stipulations for what kind of animal should be sacrificed, what colour it should have and what sex. Generally speaking, female animals were offered to female deities and male ones to male deities.
An offering rite has three steps:
praefatio
A prayer is recited and wine and incense are burned.
immolatio (letâs assume this is a sacrum publicum)
There are two options: in the ritus Graecus, a line is drawn from the head of the animal to its tail with a sacrificial knife and a few hairs from its head are burned.
In the ritus Romanus, a salted spelt, the mola salsa, is sprinkled on its head.
Then the animal is killed by a slave, but itâs very important that the animal does not get nervous. Sometimes itâs stunned with a blow to the head.
litatio
The animalâs entrails have to be inspected for any flaws or abnormalities. If there are any, the offering has not been accepted has to be repeated ad usque litationem, until the gods accept it. When itâs been accepted, the entrails are burned and therefore offered to the gods.
The rest of the meat is either cooked and served to the sacrificial community, or given to the community to carry home, or it can be sold to butcheries, where people who werenât at the offering can buy it so that they were, in a way, part of the offering rite.
There are several people who can, but donât have to, be part of an offering:
victimarii - servants who are responsible for looking after the sacrificial animal
ministri - sacrificial servants
a musician who is making noise to drive bad spirits and such away
Basically every offering is made within a specific context. Ovid describes some in his fasti (a literary calendar of sorts), like the larentalia or the robigalia. Thatâs a good source for specific rites and rituals with sometimes very detailed descriptions. Of course, not everything has necessarily been exactly the way Ovid describes it.
The compitalia, for example, an annual festival that was held to honour the lares compitales, the lares of a whole village or city. People would hang as many puppets as residents lived there into the compital chapels, and little wool balls for the slaves. Then they would offer and celebrate, and the slaves would get an extraration of wine.
The cults also have rites of their own that can be quite peculiar. The cult of Isis had an annual festival where they would search for the corpse of Osiris in a body of water, âfindâ him and then have a very noisy pompa, a procession, to the sea and back to the temple, where they would offer.
The cult of Bona Dea held an annual festival called the December festival, where only women were allowed. They would gather in the house of a member of the societal and political elite and make an offering, plus celebrate, of course.
The fasti by Ovid (there should be lots of translations)
Basically any book on Roman cults, there should be at least one ritual in there
I know one or two more cults, and Iâm happy to do some research, so donât hesitate to ask (though at the moment Iâm a little busy so it might take some time).