what is left of a larger whole: the rest, the remainder, or the surviving remnant
At its core this word simply names the part of something that is left once the rest is accounted for. It is a quantitative idea before it is a theological one: the rest of the money, the rest of the people, the rest of a king's provinces, the remaining trees of a forest. In this plain use it works much like the English "the rest of" or "the remainder."
The same word takes on real weight, though, when what is left over is what survives a disaster. After an army, a city, or a nation is cut down, the piece that remains is its remnant, its survivors. The prophet Isaiah made this the heart of a hope: a remnant of Israel would come back to God even after judgment stripped the nation bare, an idea Isaiah built into the very name of one of his sons, "a remnant shall return." He also speaks of the leftover survivors of Israel's enemies, whether Babylon, Moab, or Syria, and there the shrinking remnant is a threat rather than a promise.
These two uses are not different words but two shades of one idea. The link between them is loss: a remnant is the rest of a group after part of it is gone. Whether the tone is neutral bookkeeping or the drama of survival depends entirely on the setting, which is why translators move between "rest," "remainder," and "remnant" from verse to verse.