Many hosting companies say you need to transfer your domain registration
to them when using their hosting services. Unless they
do say so in their
legal fine prints, you
don't need to transfer your registration to the other
provider.
The only times you may need to transfer your registration are:
1. Your current domain provider doesn't have what you want. Or,
2. Your current domain provider has made managing your domain name a
complete nightmare.
As Lynn said, all you need is to change the DNS or "nameservers" to that
of your hosting company inside your domain's control panel. Once done, it
will take around 24-48 hours for the changes to take effect throughout the
Internet.
Despite advances in technology, some things don't happen immediately as
we'd like to believe. This isn't due to slow technological development, but
rather due to demand growing faster than supply is able to cope with.
Personally speaking, I've seen a domain name resolve to its website in
20 seconds flat after the DNS change. But it didn't happen for another one
even though they're in the same place.
Anyway, keep on trying until your domain name finally resolves to its site.
If it hasn't after 24-48 hours, then it's time to start checking.
BTW, there
is a "rule" where domain name registrations can be transferred
to another provider 60 days after its creation date. But note this applies to
domain name registrations, not switching web hosting providers.
Hoping this helps.