An SSL certificate serves two purposes: traffic encryption (for RSA key exchange, at least) and trust verification. As you know, you can encrypt traffic (or without, if we say SSL 3.0 or TLS) any self-signed certificate. But trust is exercised through a chain of certificates. I do not know you, but I trust verisign (or at least Microsoft, because they paid a lot of money to install it on their operating systems by default), and since Verisign trusts you, I trust you too. As a result, there is no scary warning when I go to such an SSL page in my web browser, because someone I trust said that you are who you are.
As a rule, the more expensive the certificate, the larger the investigation of what issues the certificate. Thus, for advanced verification certificates, applicants must submit more documents to prove that they are who they say, and in return they get a bright, happy green bar in modern web browsers (I think Safari does nothing with it quite yet).
Finally, some companies come with big boys, such as Verisign, solely for the brand name; they know that their customers have at least heard of Verisign, and therefore for people shopping in their online store, their print looks a little smaller than a sketch than, say, GoDaddy's.
If branding is not important to you or your site is not prone to phishing attacks, then the cheapest SSL certificate you can buy, which is installed by root in most web browsers by default, will be great. Typically, the only check is that you should be able to reply to the email sent to the DNS administrative contact, thereby “proving” that you “own” this domain name.
You can use these cheap o certificates on servers other than GoDaddy, of course, but you may have to install an intermediate certificate on the server first. This is the certificate that lies between your cheap-o $ 30 certificate and GoDaddy's "real deal" root certificate. The web browsers visiting your site will look like "hmm, it looks like it was signed with an intermediary, do you have it?" for which it may require an extra ride. But then he will request an intermediate from his server, make sure that it is tied to a trusted root certificate that he knows about, and there are no problems.
But if you are not allowed to install an intermediate link on your server (for example, in shared hosting mode), you are out of luck. This is why most people say that GoDaddy certificates cannot be used on servers without GoDaddy. Not true, but enough for many scenarios.
(At work, we use the Comodo certificate for our online store and the cheapo $ 30 GoDaddy cert to provide internal database connectivity.)
Edited in italics to reflect erickson's clarifications below. Learn something new every day!