BJT saturation region
The name saturation has more to do with the bias voltage-current relationship rather than the behavior of the carriers themselves. 

In forward-active mode, the collector-emitter current is a result of the the combined action of the following:


Base-Emitter forward voltage, which causes current flow according to the working principles of a PN junction
Base-Collector reverse voltage, which results in a depletion region in the thin base region that can then capture carriers.


This second point is important: the base-collector voltage must be sufficiently reverse-biased such that the formation of a wide enough depletion region in the thin base region is possible.  As the magnitude of this reverse-bias voltage decreases, the depletion region width also decreases, and eventually the transistor stops behaving as described in forward active mode.  At this point, the transistor is said to be in saturation, because increases in base current \$ I_B \$ are not reflected as collector current \$ I_C = \beta I_B \$ .  In other words, the collector current has saturated at some value below \$ \beta I_B \$.  

Some notes about transistor operation in saturation:


Since the BJT is a nonlinear device, it is hard to pinpoint an exact voltage that corresponds to saturation mode operation as opposed to active mode.  Therefore, circuit analysis typically involves assigning a saturation collector-emitter voltage, \$ V_{CEsat} \$, below which the device is said to be operating in saturation and above which the device is said to be operating in forward-active mode.  This saturation voltage is typically something like 0.2V.  The transition of the device from forward-active to saturation is made clear in the following diagram from https://www.electronics-tutorials.ws/transistor/tran_4.html on the I-V characteristics of BJTs.




The definition of a discrete saturation threshold voltage is called "piecewise-linear analysis," and is usually a great first pass tool in circuit analysis or design.  


The piecewise-linear model described above does not work well when the BJT is on the edge of saturation (i.e. when the device is operating very close to \$ V_{CE} = V_{CEsat} \$).  This is because the piecewise-linear model is only an approximation of the device behavior.  
Saturation mode is not "good" or "bad"...designers might term saturation region as desirable or undesirable depending on the design case.  When designing a small-signal amplifier, the designer tries to keep the device in forward-active mode to exploit the high base-collector current gain, which in turn means avoiding large collector loading such that \$ V_{CE} \$ is pushed to its saturation value. However, when using the BJT as a switch, saturation mode operation works, since all you care about is toggling the device between cutoff (device is not conducting) and some conducting state with the application of a base current.  An example of this might be to use an NPN BJT to drive an LED or small motor from a microcontroller GPIO pin (check out this simplified application explanation from sparkfun for an example: https://learn.sparkfun.com/tutorials/transistors/all)