In larger Roman towns, people often got sick or died from drinking water that had been contaminated with sewage. Sewage is human waste - poop or pee. When people drink water with poop in it, they can get other people's germs and get sick with dysentery or die. To fix this problem, many Roman towns built aqueducts to bring in fresh water from the hills outside of the towns. They also built public latrines and systems of sewage pipes to carry sewage out of the streets and dump it into the river. This was a big improvement on Greek sewage arrangements, where people just poured their waste into the street however they wanted.
Public latrine at Ostia (Thanks to Lynn Lichtenbaum)
There were also big public toilets that had room for lots of people at the same time (this one is from Ostia). Unlike latrines in China, in Rome people sat down to go to the bathroom.
Street drain in a Roman street
These sewers just dumped raw sewage into the river, which was better than leaving it lying around in the streets, but still did spread germs sometimes. The Romans didn't have any way of treating sewage to kill the germs, as we do today, and they didn't understand the need to do that.
Roman clay drainpipe
Some of these sewers are still being used today.
In smaller towns, though - most of the towns in the Roman Empire - there weren't any sewers, and sewage collectors came through and got the poop from each house and carried it off to sell to farmers to use as fertilizer on their fields, just like in ancient China at the same time. In small villages (where most people lived), they didn't even have outhouses, and people just walked out to the fields every morning and pooped there.
To find out more about Roman sewage, check out this book from Amazon.com or from your local library:
City, by David Macaulay (1983). A detailed description, lavishly illustrated, of the construction of a Roman town. Incredibly accurate. Highly recommended for anyone young and interested in ancient Rome.