Poor Romans (which was most people) ate sitting at a wooden table, the way you do, on wooden stools (Most people didn’t have chairs with backs). Or, if they were slaves, they just sat on the floor or on a straw mat.
Rich people, at least when they had company, ate in a more formal way, lying down on couches around a table. Because there were usually three couches in a room, these formal dining rooms were called triclinia (tri = three, and cline = down). (One triclinium, two triclinia).
Ancient Roman Homes, by Brian Williams (2002). Easy reading.
A Roman Villa: Inside Story, by Jacqueline Morley (American edition 1992). For kids, with lots of pictures.
Ancient Rome (Eyewitness Books), by Simon James (2004). Also for kids, with lots of great photographs.
The Roman Banquet : Images of Conviviality, by Katherine Dunbabin (2004). By a specialist, for interested adults. What Roman dinner parties were like, and how they were different from Greek ones.
Houses and Society in Pompeii and Herculaneum, by Andrew Wallace-Hadrill (1996). By a leading expert in ancient architecture.