A word in Ancient Greek with the origin of "that which is heard" and literally means "fame in epic song" or "fame sung far and wide," although it is more loosely translated as simply fame, glory, or reputation.
The concept of kleos is related to the tradition in Ancient Greece of the bard as entertainment. Tales were told via oral tradition, and news and history spread through songs which were sung by the bard as after-dinner entertainment.
Homer's the Odyssey is a story about the kleos of survival and coming home--that is, earning kleos in a non-war situation. Odysseus earns kleos through nostos (homecoming) after the Trojan War. Odysseus also earns kleos by being a just king and benefitting others in his kingdom of Ithaca.
The idea of kleos is even more prevalant in Homer's Iliad. Agamemnon and Achilles discuss different ways of dying or being laid to rest, where kleos is from death and funeral. Eventually Agamemnon, murdered by his wife Klytemnestra and her lover Aigisthos, has no funeral, thus no kleos; Achilles, on the other hand, has the greatest funeral because of his heroic death in battle, and this earns him the most kleos.
Kleos is the driving motivation behind the heroic deeds of the great heroes. In a sense, it is a means to achieve an immortality, a lasting impression. Achilles, the ultimate embodiment of heroism, is given a choice before entering battle.
For my mother Thetis the goddess of the silver feet tells me
I carry two sorts of destiny toward the day of my death. Either,
if I stay here and fight beside the city of the Trojans,
my return home is gone, but my glory shall be everlasting;
but if I return home to the beloved land of my fathers,
the excellence of my glory is gone, but there will be a long life
left for me, and my end in death will not come to me quickly. -- Iliad, IX.411-7 (tr. Lattimore)
As we know, Achilles chooses the first of the two. He dies valiantly in battle in the Trojan War, and his memory lives on.