Symptoms
After updating a domain's DNS records, the domain still resolves to the old IP.
Explanation
DNS propagation is the process of changes to DNS making their way to DNS resolvers throughout the world. When domain lookups occur, they go first to a DNS resolver (typically at the ISP). If this DNS resolver has the DNS record for your domain cached, it will serve that cached record. A resolver will cache a record based on the TTL (time to live) set in the DNS record. If the DNS record cache has expired(exceeded the TTL) on that DNS resolver, it will fetch the current DNS record from your domain's nameservers, then serve and cache that record. This caching system results in some systems getting the old DNS record and some getting the new DNS record, depending on how recently the DNS resolver they are using has updated its cache. This time between changing the DNS record and when the DNS record is cached on every DNS resolver is called DNS propagation.
Please note, DNS resolvers do not need to respect the TTL and may default to a higher TTL if set low.