These are terms used by medieval philosophers, and also by Descartes and some others. Today, only a few scholars in the Aristotelian or Thomistic tradition continue to use them.
The idea is to distinguish different kinds of cause and effect relation, depending on the nature of the causing agent and its relation to the effect. If the cause is not of the same nature as the effect, it is a virtual cause. If the cause is of the same nature as the effect, it is a formal cause. If the cause is more perfect then the effect, it is an eminent cause.
If I punch you and give you a scar, the scar was never in me, so the cause only virtually contains its effect. But if I give you a sandwich, I owned or 'contained' the sandwich and now you do, so this is an example of a cause that formally contains its effect. While I formally contained the sandwich, I am not the eminent cause of it, because I do not possess an unlimited capacity to manufacture sandwiches without loss to myself. If I have a contagious disease and I give it to you, we might say that I am an eminent cause of your disease, since I still have the disease in an undiminished form and I can continue to communicate it to others. According to the scholastics, God is the eminent cause of all creation because God contains all its perfections and more, and God is not diminished in any way by creating things.
The distinction is rather obsolete and doesn't relate to how we think of causes and effects in modern science. It depends on the idea that in a cause-effect relation, the cause is an agent that is passing something to its effect.
If you are interested in writers who do make use of the distinction...
Descartes makes use of the distinction in the Meditations, particularly the Third, when claiming that a cause must have at least as much reality as its effect.
Leibniz makes some reference to eminent causes, see, e.g. the section on Divine Causation in the Stanford Encyclopedia.
Edward Feser is an example of a modern writer in the Aristotelian tradition who makes use of the distinction in his book, Five Proofs of the Existence of God.