To put it simply, DHCP Option 82 is the "DHCP Relay Agent Information Option". Wasn't that easy?
Option 82 was designed to allow a DHCP Relay Agent to insert circuit specific information into a request that is being forwarded to a DHCP server. Specifically the option works by setting two sub-options: Circuit ID and Remote ID.
The Circuit ID sub-option is supposed to include information specific to which circuit the request came in on. It's an identifier that is specific to the relay agent, so what kind of circuit is described will vary depending on the relay agent. In an Ethernet-based network this is probably a port on a switch. In other networks, this might be a wireless modem or an ATM virtual circuit.
The Remote ID sub-option was designed to carry information relating to the remote host end of the circuit. In practice, this sub-option usually contains information that identifies the relay agent. In the case of an Ethernet network, this is the MAC address of the relay agent. In a wireless network, this would likely be a unique identifier of the wireless access point. It's worth noting that the Remote ID sub-option was designed to be globally unique. In other words, Remote ID "7" should not have different meanings depending on some other context.
In it's default configuration, the DHCP Relay Agent Information Option passes along port and agent information to a central DHCP server. It is useful in statistical analysis, as well as, indicating where an assigned IP address physically connects to the network. It may also be used to make DHCP decisions based on where the request is coming from or even which user is making the request.