The terms "DCM" and "CCM" are general terms that apply to all power converters, and these refer to whether or not the inductor current remains non-zero throughout the switching cycle. If the current is at zero for a part of the cycle, then it is DCM; and CCM otherwise. The case where the current momentarily touches zero is called "BCM", "Boundary Conduction Mode" since it is the boundary case between DCM and CCM.
The case of the fly-back is a little bit special, and serves to illustrate an important point regarding coupled inductors (thanks to @Andy aka for reminding me in the comments). Let's assume we have a simple flyback with just two windings, a primary and a secondary. If we observe the current in the transformer primary winding, we see that it always goes to zero during the switching cycle regardless of whether the flyback is in CCM or DCM mode. So how do we determine if the flyback is operating in CCM or DCM? Well, that depends on what the current in the secondary winding is doing at the time the primary switch is turned on again at the start of the next cycle. If it is still conducting, then the flyback is in CCM; but if it has stopped conducting, then it is DCM. This leads us to make a more general statement about CCM vs DCM:
Consider one complete switching cycle of the converter. Now consider a plot of the energy stored in the magnetic field of the main magnetic component vs time over one complete cycle. If that plot shows a distinct period when energy is zero, then the converter is in DCM. If the plot energy is always non-zero, then we have CCM. The boundary between these two cases is BCM.
When considering whether a converter is in CCM or DCM, be aware of any coupled inductors - there are other types of converters apart from the flyback that use coupled inductors, and it can be confusing to determine if they operate in CCM or DCM.
There are already several posts on this website that will answer most if not all the questions you have. Below is a small sample, may I suggest reading these, then perhaps re-consider your questions if you need further insight.
Hope this helps.
CCM, DCM and BCM - Inductor?
Flyback converter DCM and CCM transition voltage?
CCM flyback or DCM flyback?
How to know if a flyback converter is working in CCM or DCM