Many schools and businesses use a proxy server for Internet access.
This allows them to filter certain sites and types of content that they don't want students or employees to access.
The system is able to tell what sites a specific user was on and for how long.
When you connect to a computer system via a wireless router (WiFi) your device's MAC address is stored.
Every device that has a network interface has a unique MAC (Media Access Control) address. This is usually expressed in hexadecimal such as 10:25:A4:59:EF:1C http://en.wikipedia.org/wiki/MAC_address MAC addresses are permanently assigned when the device is manufactured and can not be physically changed.
The wireless router then assigns your device a unique IP address via its DHCP service. http://en.wikipedia.org/wiki/Ip_address IP addresses are usually automatically assigned by the DHCP service running on your router, but can be set manually. They are in binary but usually expressed in readable format such as 192.168.1.1
Now you know the technical aspects of how your specific device is identifiable, I will proceed.
Every time you go to a web page, your Internet browser requests that content from a remote server be delivered to your device via HTTP.
All of the web content including the source code, formatting (CSS), graphics, flash content, etc. must be requested via the wireless router's connection to the ISP's router.
The same thing happens when you try and play an online game. Your device must request access to a specific game server via the wireless router. The router connects to a proxy server which can filter and or limit and log this activity.
In addition the wireless router also keeps detailed logs, If the owner is tech savvy, they can easily see exactly where you have been and what you downloaded by examining the wireless router log. By examining the URL's requested by your device the history of sites that you visited and files that you downloaded can be easily reconstructed.
Good Luck...