Two different roles.
AUTHORITATIVE
Servers holding the actual records for a domain. They are the source of truth.
Your domain's nameservers are authoritative for your domain.
RECURSIVE
Servers that look answers up on behalf of a device, following the chain from root to authoritative, and caching the result.
Your internet provider runs one. Public resolvers are also recursive.
WHY THE DISTINCTION MATTERS
When you change a record, the authoritative server has the new value immediately.
Recursive resolvers continue serving their cached copy until it expires.
That gap is what people call propagation.
CHECKING AUTHORITATIVELY
A lookup tool can query your authoritative nameservers directly, bypassing caches.
If the authoritative answer is correct, the change is made and you are waiting for caches.
If it is wrong, the change did not take effect and you should check where you made it.
That single check separates "waiting" from "broken", and it saves considerable time.
WHAT THIS TELLS YOU
Always check the authoritative answer before concluding a change failed.
