The amount of flights that can take place at an airport over a given period of time is limited, mostly due to the amount, placement, type and size of the available runways and taxiways. This limit is represented by the amount of Slots an airport offers per every five minute time window.
Every flight you assign to an aircraft requires an arrival and a departure slot. Slots are granted to airlines on a first-come, first-served basis. If all slots are occupied by other airlines for a given departure or arrival time, this will display as an error in the flight planning interface.
Note that a slot is immediately freed and available to other airlines as soon as you remove a flight from a flight plan or you save a flight with errors. To avoid this problem, especially when transferring the complete flight plan of an aircraft, consider using the Flight Plan Transfer Tool.
If no slots are available for a given departure or arrival time, you can try to use departure offsets and speed overrides to modify the times and possibly hit a free slot.