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After I have found this website http://www.desainew.com/2018/01/business-card-mockup-tutorial-in-blender.html it was quite easy for me to generate something similar with my own business card design.
However, it is quite flat. And it would be really cool to have something more like this:
Is it possible to give the a plane with a texture such a "pressed effect"?
There are many ways to do this.
Using an image as texture to control the displacement you can connect the texture to the Hieght socket of a Vector>Bump node, and plug that to the Normal socket of the shader you are using for the card.
This will create a bump map, faking some relief and shading the object using the existing lights in the scene.
Alternatively, you can use the texture and plug it to the displacement socket of the Material output. Fine tuning amount of displacement can be done with a Converter>Math node.
Note that with both of these node setups you will be creating the illusion of displacement without adding any geometry. But if you look at the card from a side it will look like the flat plane it is.
To get real 3D displacement, set the displacement for the material to True or Both.
For version 2.79 enable the experimental Feature Set.
Then subdivide the original mesh so that you have denser geometry.
Another alternative is to use a displace modifier that uses the texture to displace the geometry of the card.
Note that the last two techniques require more geometry, to get a smooth result requires the object to have more vertices.
Another way is to add Text, knife-project it onto the card and extrude it. You may also bevel it. And bump mapping requires a greyscale height map.
