In more detail
A DNS TTL of 300 lets resolvers cache an answer for five minutes; common values are 300, 3600 (one hour) and 86400 (one day). The domain's authoritative server sets it, and each resolver counts it down from the moment it cached the answer. Short TTLs make changes take effect quickly but increase lookups; long TTLs are efficient but slow to change. In IP packets the TTL field means something different: a hop counter that drops by one at every router and discards the packet at zero, preventing endless loops. traceroute uses this, sending packets with TTL 1, 2, 3… so each router in turn reports itself.
Why it matters
What is called "DNS propagation" is really caches expiring. Lower a record's TTL at least one old TTL period before changing it, and the switch then completes within the new, short TTL.
Example
A record has TTL 86400. Lower it to 300 at least 24 hours before a server move; after the change, every resolver has the new address within five minutes. For IP: Linux and macOS usually start packets at TTL 64 and Windows at 128, so a ping reply showing ttl=57 from a Linux host suggests 64 − 57 = 7 router hops.