Inputs and Outputs (I/O) for a PLC can be divided into two categories. Most I/O has two discrete states - on or off. These are referred to as discrete (or somtimes digital from the digits 1 and 0 used to represent the ON and OFF states). This works well for using the PLC to "see" (inputs) or control (outputs) if a motor is on or off, a valve is open or close, a thermostat is on or off, etc.
However, many real world situations involve physical quantities that vary in magnitude, such as temperature or pressure (sample inputs) and speed or position (sample outputs). In order to work with these physical measurements the PLC has to be able to sense or control a signal where the magnitude of the signal is proportional to the measuremnt or controlled variable. This is what analog I/O does. (The term probably goes back to analog computers where the DC voltage was analogous to the system being modeled.)
With an anaolg input, for example, the higher the pressure a sensor sees, the higher the value of the voltage signal it sends to the PLC is. The higher the value of the signal the PLC's input "sees" the higher the numerical value the PLC stores in its memory register becomes. Once the value representing the measured physical quantity is stored in the PLC's memory it can be used for alarming, or control, or display, or all kinds of calcultions and manipulation.
Just like discrete I/O comes in various flavors like 24 VDC or 120 VAC, analog I/O can be 0-10 VDC, or 4-20 mA, or mili-volts from a thermocouple. Look at some PLC manufacturer's sites to get an idea of the wide variety available.