I updated my DNS records/DNS nameservers but they're still pointing to the old site

If you have properly updated the DNS information at your domain registrar to use new name servers, or you have changed a setting in your DNS records, but the record still appears to be pointing to the older data, you are experiencing DNS Caching.  
Changing the DNS server information at your domain registrar may not happen immediately (it depends on the domain registrar and how quickly they pass the updates on to the central DNS servers), but making an actual change to a DNS record e.g. updating the IP in an A-record, happens immediately.
However, even if all updates are set immediately, these changes have to be passed onto every other DNS server.  Most DNS servers have a cache (an internal store) where once they have requested DNS information for a domain, they store that information for a specific period of time.  The reason they store it is so that if another user wants to visit the same service it can supply the information quickly from its own internal store rather than have to request it from an external source.  The period of time they store that information for depends upon the administrator of that DNS server. It could store if for a few minutes, or several hours, but there's nothing anyone can about this time span.
The result is that even if you update your DNS records and the change is immediate, most people connecting to your domain who have visited before will be using DNS servers that have cached your domains information during a previous visit.  If that DNS server has not updated its cache then it will be providing out-of-date information. This whole process is complicated by the fact that there may be various DNS servers (each with their own cache and time-frame to update their cache) in the chain, and even your own web browser will cache DNS information for a period of time.
Nowadays, updated/changed DNS information usually travels accross the internet (propogates) very quickly, often within an hour or two. But sometimes it can take up to 72 hours for ALL the DNS servers in the chain to be updated. There's nothing that can be done to speed up that process and you simply have to have patience.
If you think you may have made a mistake with your DNS settings, you can use services like http://www.intodns.com to check your DNS records.  But, if you check them first, then make a DNS change, the chances are your information will be cached by them too for a short period and they may not display the full up-to-date information.
The DNS caching period is a normal occurance and unavoidable.