Have a web site at home, and when I go to buy a domain for it it wants
to know my DNS servers.  What, exactly, are DNS servers?  How can I
get them for my domain?  Are there companies that will give you free
DNS servers? What are the best (free and paid)?

Hello jaggy-ga
.
DNS = Domain Name Servers, they are used to host your domain name.
It is strange that the source you are purchasing your name does not
offer domain name parking at the very least - so the name is using
their DNS info. Domain name parking is when the domain name is
'parked' on a web page that says something like "This domain name has
been registered" These days most domain name registration services
include not only domain name parking, but email services and web
forwarding to your website address.
.
I assume that the company you are referring to do have email services
and web forwarding included but perhaps the registration forms are
confusing. (It is rare for a company to only sell you a name) Usually
you have the option (not compulsory) to input the DNS details if you
have an existing website or intend to use the name with a hosting
provider of webspace - this optional input would be your web hosting
company's DNS details.
.
Anyhow a company like Nameroute.com give you a .com name for $15 a
year and includes 10mb of webspace, POP3 email, webmail, mail
forwarding, anti spam filters, and a web building guide. (they provide
the DNS server)
http://www.nameroute.com/
.
Domain-parking.net also provide the DNS server and offer you email
forwarding, webforwarding with the added functionality of IP
forwarding to an IP number and creating subdomains e.g.
abc.youdomain.com and xyz.yourdomain.com $44.07 for 2 years for a .com
http://www.domain-parking.net
.
If you find yourself in the unfortunate situation of having registered
a name without any DNS provision or with a limited feature set, 'Zone
Edit' ( http://zoneedit.com ) provide comprehensive domain name
management features but are perhaps a bit too complex for the
beginner. However it is free for the first 5 names you host on with
their DNS.
So you can buy your domain name from elsewhere and input their DNS
details like ns1.zoneedit.com and ns2.zoneedit.com (usually 2
nameservers are used incase one fails).
.
I hope that helps
regards lot-ga

Other sources of free DNS servers:
Granite Canyon operates a public DNS service
(http://www.granitecanyon.com/)
Nominum operates a free secondary DNS service for 5 or less domains
with less than 100 resource records in total
(http://www.secondary.com/)

So what you could do is to use granite canyon for primary and
secondary DNS, and on top of that, secondary.com's secondary DNS to
ensure geographical and topological redundancy for your domain

Good paid DNS services:
UltraDNS's managed DNS service (http://www.ultradns.com/)
Nominum's DNS hosting service (http://www.nominum.com/)

There's also http://mydomain.com/, who offer free, easy to use, and
fairly advanced DNS services.