Last Alliance won the battle at the gates of Mordor, entered, and besieged the Dark Tower. Trying to break the siege, Sauron himself came out and gave battle. Stones from the DarkTower killed Anárion, and Elendil & Gil-galad threw down Sauron, killing his physical body. Gil-galad died of his wounds, and Isildur cut the OneRing from Sauron's finger.
He should have destroyed it, but few saw him cut the Ring or realize what it meant. Elrond, though present, didn't notice.
Later, while the army recovered in Mordor (and while MountDoom was still close by), Elrond and Círdan counciled Isildur to destroy the Ring. Isildur claimed it as an heirloom of his house, in memory of his father and brother.
As Sauron's armies dispersed, one of them hid in the eaves of Greenwood the Great. Scouts of King Oropher of the Woodland Realm did not detect them when his army returned North.
Isildur spent nearly 3 years in Gondor, ordering its new government. Then he departed for Arnor, taking only a small riding company as escort. The Orcs attacked, and surrounded the company above Anduin's floodplain.
Isildur's eldest son Elendur advised him to the Ring on and use it against the Orcs, but he refused (possibly because Elrond warned him not to use the Ring's powers). But as the guards were all slain Isildur finally relented to use the Ring to escape. He put it on, and gave out a great cry of pain, as it was still hot. Then he fled the scene of the disaster, as the last of his guards, and his elder sons, were killed.
Isildur ran down into the floodplain of the Anduin, came to the river, and tried to swim across. As he neared the West bank the Ring slipped off his hand. As he staggered out of the water a small company of Orcs saw him rise up before them, and they unwisely shot him out of fear. He fell back into the water, and there his bones lay until Saruman may have found them, millenia later.
The Ring was now on the riverbed, in pools near the West bank, where the happless Deagol would find it, long before Saruman came looking.