An understudy typically has another role in a show, like an ensemble track. However, a standby only covers a particular role in a show and spends the performance hanging around the theatre on the off chance they are needed. (They get paid for this and are able to leave the theatre to run an errand provided that they are reachable in case of emergency.)
A more well known story of a standby going on in the middle of the show is when Shoshana Bean went on for Idina Menzel. Menzel injured herself towards the end of her second to last performance and the show was halted and Bean jumped into the show at a moments notice.
"If you try to shag my husband while I am still alive, I will shove the art of motorcycle maintenance up your rancid little Cu**. That's a good dear"
Tom Stoppard's Rock N Roll