Plumber comes from the Latin plumbum, for “lead.” A plumber originally meant someone who works with (particularly smelts) lead. These men, over time, worked mostly with pipes (made of lead!) and eventually dealt more and more with the pipes that carry water into (or out of) homes and buildings. So lead workers became plumbers.
Interestingly, the Spanish for “lead” (plomo) comes from the same Latin root. We can see the pl-m root in words in both languages.