A chat server is a computer dedicated to providing the processing power to handle and maintain chatting and it users. For example, there are thousands of dedicated servers setup for IRC, each of these servers are considered a chat server.
A chat is a system that allows users to communicate in real time using easily accessible web interfaces. It is a type of internet online chat distinguished by its simplicity and accessibility to users who do not wish to take the time to install and learn to use specialized chat software. This trait allows users instantaneous access and only a web browser is required to chat. Users will always get the latest version of a chat service because no software installation or updates are required.