During the time of the Republic, for their afternoon and evening meals, Romans ate mostly vegetables and dined very simply.
The women of the house, or the slaves under their direction, would prepare the meals, which were then served by the children of the house.
In the atrium, a dining table would be set up for meals and the family would sit on stools around it. After serving the adults the children would serve themselves and take their seats with the rest of the family. The whole family ate together.
The Romans did not have forks. They did use spoons and knives while eating. Mostly food was eaten using your hands or a spoon.
Over time for the Patrician class this simple style of living changed. Romans no longer sat at a table, instead they lay down on special dining couches. Food was still mostly eaten with your fingers.
Patricians would frequently throw elaborate dinner parties. In the early empire period these were separated into men's parties and women's parties. The two did not mix. A dinner might last several hours and usually included some type of entertainment. They still used dining couches and ate with their hands. During the later empire period, the family again dined together.
Plebeians might have a dinner of porridge made of vegetables, or, when they could afford it, fish, bread, olives, and wine, and meat on occasion. The really poor made do with whatever they could buy or whatever the government gave them.