Lord Argoz was one of the Seven lost Lords, and a Telmarine lord who opposed Miraz. These lords did not fear the oceans or forests, unlike most Telmarines, and were one of the few who respected the legends of Old Narnia, or Narnia before the invasion from Telmar.
Argoz and six other lords were sent to "explore the seas beyond the Lone Islands" by King Miraz in an attempt to get rid of them. Argoz was one of the last three lords to be on the voyage, then later fell into a deep, dream-free sleep on Ramandu's Island.
Apparently, when the Lords came to Ramandu's Island, the three remaining Lords got into a fight, unable to decide what their next course of action should be. One of them wanted to remain on the island for the rest of their lives, another wanted to return to Narnia, and the final one wanted to continue exploring. When none could reach a compromise, the third one, in a fit of anger, grabbed the Stone Knife, and in turn cursed him and his comrades to a long sleep.
It was only thanks to Caspian and the crew of the Dawn Treader that made it to the end of the world that they were able to finally awake.
In the 2010 film version, Lord Argoz had fled to the Lone Islands to escape Miraz, along with his 6 fellow Lords. Once there, they learned of the Green Mist that was abducting people. They all swore a pact to find the source of the Mist, and destroy it.
Lords Argoz, Revilian and Mavramorn made it as far as Ramandu's Island, but by the time they reached it, they were half-mad, most likely due to the Mist, or perhaps by starvation and thirst. They threatened violence upon each other, and since violence is forbidden at Aslan's Table, they were put to sleep.
Once Caspian and the crew of the Dawn Treader had defeated and vanquished the Green Mist, by laying all seven of the Seven Swords on Aslan's Table, the Lords all awoke.