Context

We appreciate that not everyone is going to want to deep-dive into research to understand a throwaway line, so here's some context for various things thrown into the show for flavour: 
 

Mort Safe 
The show's namesake!
These were metal or stone casings that coffins would be placed in in graveyards. They were hard to break into, so deterred graverobbers. After a few days (when the bodies were no longer, erm, desirable) the coffin would be removed and buried and the mort safe reused. 
None are actually used in the show, but it's a fun name, and thematically appropriate. 
 

Greyfriars 
A kirkyard (Church + Graveyard) in central Edinburgh, that buried an estimated 100,000-200,000 bodies from 1562-1870s.
 

Candlemaker Row 
The street that runs alongside Greyfriars up from Grassmarket. The city demanded the candlemaking trade move there in 1722 as to contain the rancid smell the process created.
 

Resurrectionist 
Another term for a bodysnatcher.
 

Burke and Hare
Two famous Irish serial killers who sold bodies of their victims to the medical school's Dr Robert Knox.
 

The Vaults 
When the South Bridge was built in 1788, the inside was left hollow as to create storage rooms and workspaces for the businesses above. Unfortunately, these were prone to damp, and by 1800 it had become a hub of illegal brewing, gambling and prostitution, and an extension of the Cowgate area's slums.