Any time that DNS changes are made (on any level), you need to wait for propagation to complete. Propagation may take 24-48 hours for changes to update throughout the Internet. In some cases the change could happen nearly immediately and other cases much longer - depends on the cache refresh rate. During propagation, traffic may come to either location. One person may see the new server while someone else sees the old one. Also, yourdomain.name may work while www.yourdomain.name does not. All of this is normal during propagation.
Please note that this means that just because you go to the new site when you type in the domain name does not mean that propagation is complete. The best way to determine whether or not propagation is complete is to review the statistics for each site (the old one & the new one).
Detailed explanation of propagation:
DNS stands for Domain Name Service. Every time you go to a web site using a domain name, you are using DNS. Your request for thatdomain.name goes to your local primary or secondary DNS server (which is usually administered by your ISP). Your local DNS server checks its records to see if it "knows" what IP address that domain points to. If it does, then it directs you to that IP address. If it does not, then it sends a query to the Root DNS servers.
The Root servers are what makes the process work. When a domain is registered, it is added to the Root servers. When a domain is expired, it is removed from the Root servers. The Root servers tell your local DNS server which DNS servers ("out there") are Authorative (are the primary & secondary DNS servers) for your domain.
Your local DNS server then queries the Authorative DNS server, and the Authorative DNS server tells your local DNS server what IP address the domain is located at.
Your local DNS server then caches (makes a copy of) this information. This caching (copying) process is essential: not only does it speed up future queries, it also reduces load on the Root servers. It is this copying (caching) process that leads to propagation.
Your local DNS server does not keep that information forever. Instead, it keeps the information for a certain amount of time, at which point it deletes it. The next time (after the information is deleted) that you try to visit that domain, the process starts all over again.