Glue records
Glue records exist to solve one circular problem: finding a nameserver whose own address can only be looked up from the zone it serves.
The circular problem
Suppose example.com is served by ns1.example.com. To find example.com a resolver must ask ns1.example.com, but to find ns1.example.com it must ask the nameservers for example.com. Neither step can complete.
Glue records break the loop. The parent zone publishes the address of ns1.example.com alongside the delegation, so a resolver receives the address in the same response that tells it where to go.
When glue is needed
Glue is required only when a nameserver name sits inside the zone it serves — a configuration usually called in-bailiwick. It is set up through the registrar, often as a "host record" or "child nameserver".
If your nameservers are named under someone else's domain, as they are with most managed DNS providers, no glue is needed: their addresses resolve independently.
Stale glue
Glue is served by the parent zone, so changing the address record inside your own zone does not update it. A nameserver that moves address needs its glue updated at the registrar as well.
Stale glue produces resolution that works from some resolvers and not others, depending on whether they used the glue or looked the name up separately.