The easternmost and shortest of the many rivers that flowed out of the White Mountains and across Gondor's southern lands. Erui's springs lay in the mountains above Lossarnach, some twenty miles westward of Minas Tirith. From there it flowed nearly southward across Lebennin for approximately sixty miles until it emptied into Anduin upstream from the harbour city of Pelargir.
As it passed through Lebennin, about halfway along its course, the river was crossed by the road between two of Gondor's most important cities. This was the road from the port of Pelargir, which met the river at the place known as the Crossings of Erui before continuing on to Minas Tirith, some forty miles to the north. It's not recorded whether the river was crossed by a bridge or a ford, but given the importance of this route, it seems likely that there was a bridge at the Crossings.
Erui was the first of the many rivers of Gondor, counting westward from Anduin, but while its name contains er-, referring to singularity, it explicitly does not mean 'first'. According to notes by Tolkien quoted in The Lord of the Ring: a Reader's Companion, this river gained its name because it was '...the shortest and swiftest of the Rivers of Gondor and was the only one without a tributary.'