It takes awhile for the 'old information' in the system to be replaced by the updated version.
When you try to go to your site it asks the DNS system for the nameserver (like a local phonebook) for your site. The system sees that it looked up that information 12 hours ago, so just gives you the old information, figuring that nothing has changed. After some time, the system says "Ok, I will go get updated information".
Other people can visit your site because they never requested the information and their DNS system has to start off getting a new copy of the nameserver information.