Negative resistance is a characteristic of a circuit whereby the current required (or demanded) goes up as the voltage goes down.
A common example is a PC power supply. If your mains voltage dips, the current drawn from the mains increases to maintain the same power to the load. In general this is a characteristic of a switch mode power supply.
You cannot actually buy a negative resistance resistor though
Circuits which exhibit this behaviour are generally more complex than a simple 2 terminal device.
Having said that, some semiconductors exhibit negative resistance characteristics over a small range of voltages and/or currents. The classical example is the tunnel diode.
A tunnel diode is a two terminal device that can amplify.
There's plenty of information if you Google for "negative resistance".