The short answer – but that doesn’t mean that you should stop reading – is:
Vermouth is a fortified, aromatised wine.
Essentially, you take a neutral wine (historically, a poorly made, probably oxidised, local wine), give it some flavour and then stabilize your concoction by adding brandy (or some similar distilled spirit.)
The hallmark botanical for vermouth is artemisia, otherwise known as the mystical herb, wormwood.
Wormwood in English -> Wermut in German -> Vermouth in French.