In S01E13 "Take Me Back to Hell" Lucifer almost dies.
Is it explained at some point what this exactly means in his case?
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This actually gets brought up in a later episode.
Without getting too heavy in spoilers, when a celestial dies in Lucifer they
Simply cease to exist
Evidence to this in the spoiler tags below. It spoils season 2 and beyond.
In Season 2, Episode 5, Lucifer kills his brother Uriel with Azrael's Blade. Uriel is never seen again. He is in neither Heaven nor Hell. His body becomes lifeless and he simply ceases to exist. It is shown frequently that celestials are capable of entering Heaven and Hell at will, so the fact that Uriel never shows up again means he no longer exists. Whenever he is referred to in later episodes, it is always in the passed tense confirming that he is indeed gone.
Later in Season 5, the lack of a soul and the ending of existence becomes an obsession for Mazikeen. She expresses anger toward her mother for creating her without a soul. She explicitly states that despite being built for war, if she dies she will cease to exist as she doesn't have a soul to continue existing.