I think, when I was taught mythology in school, the river that the dead was ferried across was the Styx. However I have read that in ancient Greek mythology, Acheron was known as the river of pain, and was one of the five rivers of the Greek underworld. The newly-dead would be ferried across the Acheron by Charon, to enter the Underworld. Maybe some books or people got confused since the Styx runs off of the Acheron river.
There were actually five rivers of the Underworld; Styx was the river of hate and seperated the living realm from the realm of the dead. Acheron was the river of woe, Cocytus was the river of lamentation, Phlegethon was the river of fire, and Lethe was the river of forgetfulness.
It is thought that Charon, the old ferry man who ferries the dead to the Underworld, crosses the river Styx where the three-headed dog, Cerberus guards, allowing all souls to enter but none to leave. This is a misconception, Charon crosses the river Acheron where also Cerebus stands his eternal guard. Charon only takes the souls across that are buried properly with a coin (called an obol) that was placed in their mouths upon burial.
If a god gave his oath upon the river Styx and failed to keep his word, Zeus forced that god to drink from the river itself. The water is said to be so foul that the god would lose their voice for nine years.