Analog circuits have a range of voltages, rather than just two as in digital logic.
Analog circuits work with continuously variable voltages, in contrast with a series of discrete voltage levels. Wikipedia is wrong about having to be time-continuous, though it's rare to have analog voltages in a discrete time system (a sample-and-hold is an exception).
Analog circuits are used in the processing of sensor inputs with signals proportional to a physical quantity (see sensor). The complete circuit, up to the output, may be analog, or in a mixed analog/digital circuit the signal may be converted to digital after the analog input stage.