What are the brake housing stoppers on the end of the brake housing called (see pic) My brake cable snapped while riding my (80s road) bike one day so I replaced it.

After replacing I noticed that the piece fixing the housing to the brake was missing so that when I pulled the brake lever the housing comes too and it doesn't move the brake. 

My questions: what is this piece called, and is it something I can buy separately, Is it something that comes with brake line housings, or is it part of the brake lever itself?
They're essentially part of the brake lever. They don't really have an agreed-upon name. Probably the closest is "mushroom ferrule," which appears to be what Dia-Compe calls them.

They're not totally universally sized. Different brake levers have slightly different bits that go there but do the same thing. That said, the ones on the great majority of bikes with applicable levers are all the same size, which I believe is basically Dia-Compe and clones.

You can just buy the common size ones new. In the US, J&amp;B Importers (one of the largest parts wholesalers) has a house brand one they call a "Cable End Button" and UBS has one they call "UNIV BRAKE CABLE ADAPTER." There's also Dia-Compe part 94, same thing. You can just order any of them from an online reseller, or find a shop that has them. Some shops will also have old levers around to cannibalize.