Q: Does changing nameservers turn over A and CNAME record control to web hosting? For example, if I have registered a domain with GoDaddy, and I point the nameservers for the domain to HostGator, do I then edit the DNS Zone from GoDaddy or HostGator?
I don't really understand the relationship between nameservers and A/CNAME records so forgive me if this sounds like a silly question.

A: The question, as asked does not make a lot of sense, but DNS works as follows:


*

*You have a registrar who you pay for the domain name.  This may or may not be Go Daddy or Hostgator.   

*When you register the domain name you can specify the name servers to use.  Very often the registrar will allow you to use their nameservers.  The nameservers which are specified dictate where the domain records are edited.

*When you edit the A/CNAME or other records you are editing the Zone file for the domain.

*An A record is an IP (IPV4) address.

*A CNAME is a pointer to another domain name.  There are limits to what you should do with cnames - for example, while its OK to point www.example.com to www.cbd.ef,  its a bad idea to point example.com to cbd.ef for technical reasons (which would be the subject of another post).  

*Using CNAMES gives your provider flexibility to change IP addresses without needing to talk to you (eg for load balancing, change of servers).  Using A Records is faster as there is less work in looking up the resource, and also simpler - as hinted at before, there are hidden gotchas using CNAMES.

*A Nameserver is a server which describes how Internet resources can access your domain name.  There are lots of types of records, including MX for mail, CNAMES, A Records, TXT description records, nameserver records.   If you own example.com,  your DNS server can handle (anything).example.com - including, for example, referring requests to abc.example.com and (anything).abc.example.com to another nameserver - possibly one which someone else controls.
[ There is a hierachy of domains controlled by DNS,  at the end of the process there are special nameservers called "root" nameservers, which specify the initial part of the structure - for example .com, .net, .uk etc (ie the bit from the last dot) - Each of these root zones then points to nameservers which handle that part of the DNS, and they in turn can point to other nameservers to handle/delegate parts to the left of the zone - usually deliminated by a "." character ]

A: 
For example, if I have registered a domain with GoDaddy, and I point
  the nameservers for the domain to HostGator, do I then edit the DNS
  Zone from GoDaddy or HostGator?

Depends. Either. And doesn't even have to be either of them.
It depends on who is managing DNS. This might seem confusing since hosting companies and registrars want to imply you "must" be with them for all things, the reality is there are three broad things you are dealing with.
The three big things you are dealing with even if you don't know it.


*

*Domain Registrar: They manage domain registration and hold the records to the DNS authority entries that tell the world, "You want to know who this domain is? Check these DNS records."

*DNS Management: Who manages the DNS authority servers for your domain.

*Hosting: The place that actually hosts the services tied to the IP address that is tied to the hostname that you have registered. Sometimes the control panel of a hosting service will require a domain name be entered for a site. But that is connected to the concept of virtual hosts being tired to a site and not really a DNS-centric issue.


In general, the place you register your domain name will handle DNS management. This is default practice in the industry. So in this case, GoDaddy would be the one you deal with for A records, CNAMES and the rest of the DNS alphabet soup.
But it's not always that clear cut.
But just because your domain registrar offers DNS services doesn't mean you have to use it. For example, you can register the domain with GoDaddy, then have DNS management handled my Amazon Route 53 and then have hosting handled by HostGator.
Why would you do something like that? Depends. Domain registration is fairly silly and simple, but DNS servers are a tad more critical to the whole mix. So many people prefer to have a separation between domain name registration and DNS management.
In the case of Amazon Route 53, Amazon's DNS servers are globally allocated and highly reliable, and easier to manage. Many domain registrars let you edit DNS zone files, but their user interface is horrible. In contrast, Amazon's Route 53 interface is painfully easy and robust to use. For some people the small cost of having a separate DNS management service is worth it when dealing with ease of use.
