DNS stands for the Domain Name System. The DNS records tell the internet where to send your internet traffic. The most common (that I will discuss later) are address (A) records, CNAME records, name server (NS) records and mail exchange (MX) records.
Many web developers simply tell their clients to point the Name Server Record to the new host and you let the web developer manage that for you.
So, what does this mean for you? How do you know what you should do to maintain functionality of your site and your email? Bear in mind that my descriptions below are what I tell my my clients, but I think everyone should understand these details.