Firstly make a difference between pressing and counter pressing. Counter pressing starts at the moment of losing the ball, while pressing is an organised interception starting for a given sign, when the opponent have the ball.
So let's define the attacking organization tasks: (we just need the first two tasks, so I don't write the others)
1. Allocate the ball into a safe place after interception
2. Bring out the ball
a., Incease the game space in width and in depth
Counter pressing is about to prevent the first task, pressing is about to prevent the second task.
The mental purpose of the defending is to reduce the space and the time, which need for accomplishing the activity with the ball. It has two methods: zonal marking for reducing the space, and man marking for reducing the time. In a match both of them appears, but sometimes one of them is more dominant.
The purpose of zonal marking is to delay the attack, and to force the opponent into a mistake.
The purpose of man marking is to cut down the time for the attacking player to recept the ball, and to tighten the space, which needs for transmitting or holding the ball. In a part man maring is zonal marking.
Pressing's and counter pressing's shape element is diamond. Why? Because this is the best formation to control the space. The second point of attacking organization tasks says, that the attacking team have to increase the space in width and in depth, and you can prevent this with a diamond formation, because the two players, who are located at the two sides of the diamond can deny the space increasing in width, and the bottom member of the diamond can counteract the space increasing in depth. We will see this perfectly in the examples. These examples are from Barcelona, Bayern Münich, and Athletic Bilbao matches.
As I wrote before pressing is an organised interception starting for a given sign, when the opponent have the ball. Of course we can see here the diamond formation.
So here is the diamond, the defender one (who is the nearest to the ball at pressing, now Messi) attacks the player, who has the ball, the defender twos (the two side of the diamond, now Iniesta and Pedro) protect the wings (if the next acitvity is a side pass they will be the defender ones, who have to attack the ball), and the defender three (the bottom member of the diamond, now Xavi) is about to deny the depth pass.
Other example of pressing from this match:
Barcelona's pressing is zonal marking and man marking at the same time, but sometimes one of them is more dominant. The defender one and defender three use man marking dominantly, while defender twos use zonal marking dominantly.
Some examples from Bayern Münich.
Bayern Münich's system is like Barca's system, the only difference is, that Barca uses a more agressive type of pressing.
But Athletic uses an other type of pressing. They use man marking not only at the area of pressing, but at the other parts of the pitch. Bielsa organises a man marking, so everybody have their own man, except of a central defender, who is called "spare-man".
In this picture we can see the man marking, the two player signed with a blue point are the defender threes, so that's why there are two diamonds. The "spare-man" is framed. On the next picture we can see the same (2 diamonds, 2 defender three, man marking).
Counter pressing starts at the moment of losing the ball. Examples from Bayern and Barca.
When we saw, that in pressing the defender two use zonal marking and man marking too, but in counter pressing they use man marking dominantly. The defender one and defender three have the same role.
After losing the ball, the defender one (Inista) immediately attacking the player with the ball, the two defender two is the two full-backs, who are placed in the middle of the pitch (to tighten the space), the defender three follows the player, who comes back, the result is interception.
This is a perfect example of counter pressing, and by the way there are more Man Utd players in the pictures, but because of the well organised counter pressing, that doesn't matter.