Ténèbres translates to "darkness" or "shadows". It's a more dramatic and poetic word than the common "obscurité", often carrying connotations of despair, evil, or a profound lack of light.
In this song, Aznavour uses it to express the deep emotional desolation and despair he feels after his love has left him. He sings, "Puisque mon cœur blessé se bat dans les ténèbres" (Since my wounded heart fights in the darkness), vividly portraying his internal struggle and the profound sadness that envelops him.
Aznavour turns one heartbreak into the end of the world. Sorrow has aged him overnight, carving a line into his forehead and leaving his past feeling like an old man with nothing left to say.
From there the song keeps raising the stakes: the sea should rage over the land, hell's fire should burn every civilization, the earth should crack open and swallow everything in forgetting. Each disaster is just another way of saying the same plain thing, over and over, in the refrain: his love has left with someone else.
By the end even his own body carries the wreckage, a heart with only one cry left, day and night. It's heartbreak told at apocalyptic volume, but the ache underneath all the fire and ash is simple and familiar.