Roman people played most of the different kinds of games that people play today. The most important exceptions are card games, which were not invented until the late middle ages, and chess, which did not come to Europe from India until the Islamic Empire, about 1000 AD.
The most common games were probably dice games, where you threw the dice and bet on the results. These were essentially gambling games. But the Romans also played marbles and knucklebones (the ankle bones of sheep or goats).
Knucklebones is like jacks today. You throw up one of the knucklebones and try to scoop up some of the other knucklebones and then catch the first one before it hits the ground. You can (and Roman kids did) also play with with any small stones. The Egyptians also played this game.
The Romans also played games like checkers, and a lot of games where you moved pebbles from one square to another in a grid. We find these grids scratched into floor stones and floor tiles all over the Roman Empire, in houses, and by guardhouses, and in amphitheaters, wherever men or women, boys or girls, had some time to waste.
The Romans also played more active games. They played ball games, sometimes with a small ball and sometimes with a big heavy ball that was more exercise. They skipped seashells into the ocean, the way we skip flat stones.
Even though the Romans did play a lot of games, these games did not have the same religious importance for the Romans that they did for the Greeks. To the Greeks, physical fitness was something that made the gods happy, and physical beauty was the delight of the gods, and so exercise and athletic competition were religious activities, meant to please the gods.
For the Romans, physical fitness was not so much about the gods, but more about being able to fight in a war, and so it was not as much about being graceful and beautiful as it was about being tough.
Learn by doing: play knucklebones with small stones
More about gladiatorial games
Spend the Day in Ancient Rome: Projects and Activities that Bring the Past to Life, Ages 8-12 by Linda Honan (1998). But the project to make knucklebones seems silly. Ask your butcher for real bones, use pebbles, or buy jacks. Easy reading.
Ancient Rome (Eyewitness Books), by Simon James (2004).
Kids Around the World Play!: The Best Fun and Games from Many Lands, by Arlette N. Braman (2002). Easy reading. There are lots of ancient games in here too, even though the title doesn’t say so.
Children and Childhood in Roman Italy, by Beryl Rawson (2003).