Nameservers are the systems behind DNS. Each domain name has nameservers specified in its records and when information is needed about a particular domain it is these nameservers that provide the answers.
To speed up the process there are caching nameservers. These store the answers for a period so that the same queries aren't repeatedly having to be sent to the domain's nameservers. It is these caching nameservers that give us DNS propagation - this may be a term you've heard before and is important when DNS changes are made because it can take generally 24-48 hours for the changes to filter down to all the caching nameservers in the world. Your Internet Service Provider (ISP) will use caching nameservers and as everyone uses different ISPs this explains why some people may be able to see your fancy new website while others can't - it takes time for the updated DNS information to propagate across the Internet.