The Domain Name System (DNS) is a technology for managing the names of servers connected to Internet. The real address of a server is the IP address, but since IP addresses consist of numbers only, DNS technology allows translating a name to an IP address.
We can compare it to zip codes. Every home in America has a unique extended zip code, but when you invite someone to visit you, you don't tell him your zip code, he expects a city and street address, doesn't he? Well, there exists a database that translates all addresses to zip codes and vice versa. DNS servers do that in computer world.
The information DNS servers give can be very short. For example, if you want the IP address of dnsazure.com, the answer might be something like "22.214.171.124". That's just 15 bytes. But the information is vital. Without that info, all browsers are helpless.
Every domain like dnsazure.com or google.com has an authority, a DNS server that tells what the correct and up-to-date information really is. But that does not mean that every time you look for an address like dnsazure.com, your browser turns to that authority. Most probably, it turns to a DNS server of your company or your ISP and gets the answer from there. These servers, in turn, turn to other (more important) server to keep their records up-to-date. Finally, there are thousands of DNS servers around the world that cache the short but important address-to-IP records for clients everywhere.
Therefore, every DNS records has a time-to-live (TTL) value. The cheaper the DNS service, the longer the TTL value, because that way, less queries are made to the authority. On the other hand, the longer the TTL value, the longer it takes for every change to reach to every point of the world. Say, a DNS record of your server has its TTL value set to 24 hours and you move your server and change the DNS record. Now, all DNS servers around the world have their information in memory and they don't even bother to ask if something has changed before the TTL time is up. So, your server might appear dead for 24 hours after the change.