Open-collector/open-drain devices sink (flow) current in their low voltage active (logic 0) state, or are high impedance (no current flows) in their high voltage non-active (logic 1) state. These devices usually operate with an external pull-up resistor that holds the signal line high until a device on the wire sinks enough current to pull the line low. Many devices can be attached to the signal wire. If all devices attached to the wire are in their non-active state, the pull-up will hold the wire at a high voltage. If one or more devices are in the active state, the signal wire voltage will be low.
An open-collector/open-drain signal wire can also be bi-directional. Bi-directional means that a device can both output and input a signal on the wire at the same time. In addition to controlling the state of its pin that is connected to the signal wire (active, or non-active), a device can also sense the voltage level of the signal wire. Although the output of a open-collecter/open-drain device may be in the non-active (high) state, the wire attached to the device may be in the active (low) state, due to activity of another device attached to the wire.
The bi-directional nature of an open-collector/open-drain device is what makes this circuit so important in interconnecting many devices on a common line. The I2C Bus and SMBus uses this technique for connecting up to 127 devices.
Open-drain refers to the drain terminal of a MOS FET transistor. Open-collector is the same concept on a bipolar device.

In the picture above, the transistor base is labeled "IC Output". This is an internal output from the internal IC logic to the transistor. From the point of view of the transistor, this is the input which controls the transistor switching. The external output is the transistor collector, and the transistor acts as an interface between the internal IC logic and parts external to the IC.

The output essentially acts as either an open circuit (no connection to anything) or a connection to ground. The output usually has an external pull-up resistor, which raises the output voltage when the transistor is turned off. When any transistor connected to this resistor is turned on, the output is forced to nearly 0 volts. Open-collector outputs can be useful for analog weighting, summing, limiting, etc., but such applications are not discussed here.

A three-state logic device is unlike an open collector device, because it has transistors to source and sink current in both logic states, as well as a control to turn off both transistors and isolate the output.