A myogenic heart contracts by itself without any external stimulus, while a neurogenic heart contracts after receiving an external stimulus in the form of a nerve impulse.
Myogenic is the term used for muscles or tissues that can contract on their own, without any external electrical stimulus, from the brain or spinal cord for example.
An example of this phenomena is actually present in our kidneys to regulate the flow of blood in vessels.
Another example is the human heart.
The muscles of the human heart are stimulated to contract by nerve impulses generated by the Sino Atrial(SA) node. It is a cluster of cells which are part of the heart muscle.
Hence the human heart is myogenic. It does not require nerves to start contracting, it can contract on its own . There are nerves supplied to the heart but they only change the rate of heartbeat and cannot initiate muscle contraction.
The impulse from the SA Node, which is just an electric current, then goes down a path through the heart, stimulating the contraction of each muscle in turn as shown here in dark red lines:
Neurogenic is the term used to describe a muscle or tissue that requires an external electrical stimulus to start contracting.
As an interesting side point: Crustaceans have neurogenic hearts.
Here comes our Neurogenic friend the lobster: