Secondary DNS: adding redundancy to your name servers
Kort antwoord
Secondary DNS means a second, independent name server, often from a different provider, holds a synced copy of your zone and answers queries for it too. If your primary name servers become unreachable, resolution keeps working from the secondary, so one provider having a bad day doesn't take your domain offline.
Why one set of name servers isn't enough
A domain's NS records list the name servers that are authoritative for it, and every one of those servers is expected to be able to answer for the zone at any time. If you run only one authoritative name server, or every name server you list happens to sit behind the same provider, that provider becomes a single point of failure for your entire domain. It doesn't matter how healthy the servers behind that domain are: if nothing can resolve the name in the first place, nothing behind it is reachable either.
This is a different failure mode from the one covered in How DNS delegation works, where the NS records themselves are wrong. Here the delegation is correct, the name servers listed are simply all unavailable at once.
How a secondary name server stays in sync
Secondary DNS adds one or more extra authoritative name servers that hold a copy of the same zone data as the primary, kept up to date through a zone transfer. The secondary periodically checks whether the primary's zone has changed and pulls a copy across, either the full zone (AXFR) or just the changes since last time (IXFR). Once transferred, the secondary answers queries for the zone exactly as if it were the primary, using the same records.
The secondary doesn't need to be edited directly. You keep making changes in one place, the primary, and the transfer mechanism propagates them outward. The two servers are described as primary and secondary because of this data flow, not because one is somehow less authoritative to a resolver asking a question, both are listed as NS records and both are expected to answer correctly.
Setting it up
In practice, adding secondary DNS means two things: configuring the zone transfer between primary and secondary so the copy actually stays current, and listing both providers' name servers together in the domain's NS records at the registrar. A resolver that asks any of the listed name servers should get the same answer, so from the outside the redundancy is invisible until the primary is unreachable, at which point queries simply succeed against the secondary instead.
This is a generic DNS pattern, not tied to any particular provider. You can run a secondary with a different company than the one hosting your primary DNS, which is usually the point: independent infrastructure means a problem at one provider is less likely to also affect the other.