A "swing" understudies the ensemble members. All ensemble members of their type / gender, and also others - in case of emergency. They do not perform in the show nightly. They ONLY perform when an ensemble member is unable to.
A "Partial swing" is a swing who performs nightly in the show, but understudies other ensemble members for scenes / songs in which they do not perform nightly.
Swings are very, very valuable people and some have made entire careers of working as a swing in every show they do because it is a very social skill set that requires an awful lot of smarts and organizational skills, in addition to a wide set of performance skills and an a bailing to handle pressure.
An "Understudy" is a person who understudies / "covers" a principal role. This person might be another principal, an ensemble member, or a swing, but they cover a principal. An understudy can cover up to three people.
A "Stand-by" is essentially someone who has one, singular job - understudying a principal. They wait in the wings in case the one (or two, perhaps) person they understudy falls ill or calls out. Stand-by is a negotiated title... Sometimes they are simply called an "understudy". In rare cases, an understudy might be called a "stand-by."
A "General understudy" is essentially the same is a stand-by but they can cover up to five people.