I wish it was that easy to exploit a machine with just the MAC address
The MAC address simply lists the physical address burnt into the network card/device when the device is manufactured. The MAC address should be unique and usually a manufacturer will have a MAC address prefix assigned to that company, for example Cisco is assigned: FC-FB-FB (hex) Cisco Systems. MAC addresses can be changed/spoofed though. The MAC address list is available from: http://standards.ieee.org/develop/regauth/oui/oui.txt
MAC addresses are used on the local LAN to locate the machine you would like to communicate with, in order for your computer to communicate with an IP Address of say 192.168.1.2, your computer which has an IP Address of 192.168.1.1 will perform an ARP request for the machine. Your computer will ARP broadcast (255.255.255.255) on the local LAN "Who has 192.168.1.2 tell 192.168.1.1"
All devices on the same network will receive the broadcast and the device that knows where 192.168.1.2 is located will reply: "192.168.1.2 is at 00:11:11:22:3d:11". Now that your computer knows where the device is located (the physical MAC Address) it will start communicating with that device.
If you want to communicate with another device that is not on your local LAN then your computer will still perform an ARP request (broadcast locally). The router on the local network will respond to the ARP request with the router's MAC address if the router has a route to the remote host, the router will respond with its own MAC address. Open up a command prompt or terminal and check the ARP entries that your computer knows about, type "arp -a". You should see your default router's IP Address and associated MAC address. Now that you know what the MAC address and IP Address is you still need to investigate how you can connect to that machine. NMAP is handy for this. You need to know what ports are open and what services are running in order to connect/access another computer the Operating System is also handy.
I'd ask your teacher to explain in detail how someone can get free access to your computer. Was there any mention of WiFi and MAC address restrictions?
All men by nature desire knowledge.