Flow separation happens when the pressure gradient of the airflow along the flow path becomes too steep. In subsonic flow, the oncoming air is first decelerated ahead of the wing, then swiftly accelerated when it flows around the strongly curved nose section of a wing. This acceleration is the consequence of the wing's curvature.
After this acceleration, how will the airflow behave? Will the airflow keep moving in a straight line without further acceleration or deceleration? (which implies that airflow will separate from the wing surface, creating a local vacuum) Or will it faithfully stick to the wing surface? In reality, the air settles at a compromise between the straight path and following the contour, creating decreasing pressure along a surface with increasing curvature and increasing pressure along surfaces with decreasing curvature. More precisely, it is always in an equilibrium between inertial, viscous and pressure forces.
This suction not only bends the airflow into following the wing's contour, but also accelerates the air ahead of it. The lower the pressure, the more the air speeds up, such that the total energy of air (the sum of pressure and kinetic energy) stays constant. Therefore, pressure and local speed change in sync.
When the curvature decreases further downstream, the flow path becomes straighter and pressure rises again. However, the air particles close to the wing slow down because of friction. The layer of air where this slowing is noticeable is called boundary layer. In it, the deceleration effects due to pressure rise and due to friction add up, and at some point the air comes to a standstill relative to the wing. Where this happens, static air will collect and build up, causing flow separation. Thankfully, the exchange of air across a turbulent boundary layer kicks the slowest air particles downstream, so at moderate angles of attack the air still moves along until it reaches the trailing edge. Only when the suction peak around the nose becomes very high at high angle of attack, the consequent steep pressure rise along the remaining flow path overpowers the possibilities of the turbulent boundary layer, the air decelerates completely and the flow separates. This is a wholly subsonic affair.
If the wing moves at high subsonic speed, the curvature-created suction accelerates the flow such that it reaches supersonic speed. Now something odd happens: Supersonic flow accelerates further when subsonic flow would decelerate. This is caused by the [change in density which is dominant at supersonic speed]. Incompressible (= very slow) flow has constant density, and all speed changes affect pressure. At Mach 1, the pressure and density changes are of equal magnitude, and in supersonic flow the density changes dominate. Now we have a supersonic pocket of air on the upper surface of the wing where speed increases and density decreases downstream, and the surrounding subsonic air sees little change in density. This picture should give you some idea how it looks:
The whole wing in the picture above moves at Mach 0.68. Compare the green color at some distance from the wing with the scale on the left side, which gives the Mach number for each color shade. At the airfoil nose, you see a blue area. This is where the air decelerates - it gets pushed together by the approaching wing. Now follow the colors along the upper side - they quickly turn green, yellow and red as the air is accelerated into the low pressure area (remember, low pressure equals high speed, so the reddest area has the highest local flow speed and the lowest pressure). In subsonic flow, the suction peak would be somewhere between 20% and 30% of chord, and the colors would slowly change back to yellow and green if you move further downstream. Now we have local supersonic flow (everything redder than light orange is supersonic here), and instead of slowly decelerating, the air is speeding up to a maximum Mach number of 1.23 at almost 60% of chord length.
This cannot last, and at some point this supersonic pocket collapses. This happens instantly in a shock, and as you know, in a straight shock density increases suddenly and speed decreases such that the Mach number after the shock is the inverse of the Mach number ahead of the shock. In the picture above, boundary layer effects create a lambda shock, which has its name from the Greek letter which looks like the shock pattern here. Aft of the shock, you have subsonic flow again, and a much thicker boundary layer which moves very slowly (blue shade). This is due to the energy conversion through the shock, which converts kinetic energy into heat. But the flow is still attached - even this shock did not cause separation.
If this pressure rise is big enough, the boundary layer will come to an instantaneous standstill, and then the flow separates. This is the shock induced separation you asked about. Unfortunately, the picture above is the best I could find, and I have none with separated flow aft of the shock. But it helps to show that the center of pressure moves aft. This causes a strong nose-down moment. Also, with higher subsonic Mach numbers directional stability decreases. Now even more nasty things can happen: The location of the shock might move forward and back. This changes the size of the supersonic area, causing lift changes. On a horizontal tail, this will also cause pitch changes. If you change elevator position slightly with such a shock on the horizontal tail, the lift change could be severe and in the opposite direction of what you would expect. This causes total loss of control, just when you need the control surfaces to counteract the Mach effects mentioned above. Also, the shock position could oscillate, causing a buzzing sound and, if you are really unlucky, coupling into an elastic eigenfrequency of your structure, resulting in flutter. Not only on the tail surfaces, but also on the wing, affecting the ailerons as well. Now you can start to see what scared the early pioneers about flying near Mach 1 and why they spoke of a "sound barrier".
If you fly fully supersonic, this effect goes away because now the shock moves to the trailing edge and stays there. Now everything will be calm again because the shock location stays fixed. This effect was first experienced and survived on April 9, 1945 by [Hans Guido Mutke in a Me-262], which briefly flew fully supersonic in a dive. However, even in fully supersonic flow separation is possible, but then because the air flow will not bend more than what can be caused by total vacuum. In hypersonic flow the density changes become so severe that pockets of "air" are possible which do not contain any air, but a vacuum. But this is more an academic case, except for reentry vehicles with a blunt, rear-facing base.