In commercial network routing between autonomous systemss, hot-potato routing is the practice of passing traffic off to another AS as quickly as possible, thus using their network for wide-area transit. Cold-potato routing is the opposite, where the originating AS holds onto the packet until it is as near to the destination as possible.
Hot-potato routing is a cheapskate's approach to routing, and generally results in lower quality of service. Cold-potato routing, on the other hand, is more expensive to do, but keeps the traffic under your control for longer, allowing operators of well-provisioned networks to offer a higher QoS to their customers.
The terms can also be used to describe the route announcement policy of a network: by choosing to announce their network at a large number of points at the periphery of another AS, a provider can pull incoming traffic onto their network as soon as possible, ensuring that the traffic stays on their network all the way to their customer's connection.