to be left alive after others have died: to survive, to remain over
This is a rare verb that Paul uses only twice, both times in the same passage about Christians who die before Jesus returns (1 Thessalonians 4:15, 17). It pictures the people who are still living at that moment as the ones who are "left over" after others have already gone, the survivors of a sequence in which some have died and some remain.
The word carries more weight than a flat "those who remain." In older Greek it is the word used for the few who come through a disaster alive: the lone survivor of a battle, the small remnant of humanity that lives on after a flood. That coloring of surviving a calamity is built into the word, and Paul's setting brings it out, because he sets these living survivors directly against the believers who have already died.
Paul pairs the word with a plainer phrase, "those who are alive," so the two sit side by side as near-twins. He builds up the apparent advantage of the survivors only to overturn it in the very next line: the dead in Christ rise first, and then the living are caught up together with them. The point is comfort, not ranking, so no one who has died will miss anything.
Readers should not hear the popular "left behind" idea in this word. The ones who are "left" in Paul's Greek are the saved who are still alive, not people abandoned at a secret rapture.