If an electron in a two level atomic system is excited by narrow line width coherent electro-magnetic radiation, like a laser, that is on resonance with the two level transition, the electron will Rabi flop. During Rabi flopping the electron oscillates between the ground and excited states and can be described by a continuous rotation around the Bloch sphere. 
For a perfectly isolated system the Rabi oscillation will continue indefinitely and will undergo no phase change, making it a "coherent state". In physical systems interactions between the system and the environment introduce an unknown phase in the Rabi oscillation between the two levels with respect to the Rabi oscillation in the perfectly isolated system causing "decoherence". 
If instead of a single two-level system an ensemble of identical two level systems (such as a chain of identical atoms in an ion trap) is prepared and continuously addressed with a laser, all the atoms may begin to simultaneously Rabi flop. At the beginning all two level systems will have a defined relative phase relation (they will all be in phase) and the system will be coherent. 
As atoms begin to undergo random spontaneous emission their Rabi oscillations will accumulate a random relative phase with respect to each other and become decoherent. In actual experiments ambient magnetic field noise and thermal heating from collisions between atoms cause decoherence faster than random spontaneous emission and are the dominant uncertainties when running atomic clocks or trapped ion quantum computers. Atomic coherence can also apply to multi-level systems which require more than a single laser.
Atomic coherence is essential in research on several effects, such as electromagnetically induced transparency (EIT), lasing without inversion (LWI), stimulated raman adiabatic passage (STIRAP) and nonlinear optical interaction with enhanced efficiency. 
Atomic systems demonstrating continuous superradiance exhibit long coherence time, a property shared with lasers.