The book Waves in an Impossible Sea really goes into some depth on this (for a layman -- which I am) and tries to drive home the point that there are two perspectives one might take. There's the perspective of the medium and the perspective of the field.
Using wind, as an example, we can measure the wind speed/direction at various points in a given space. We don't need to know what wind is to feel its effects. Instead, we might view it as a force wave that propagates through space and interacts with everyday objects. The measurements of this force that we take at various points in space across a given area form what we might call the Wind Field. We don't need to know the nature of the medium these wind waves propagate through in order to study wind and how it interacts with other objects. This is the field perspective.
Of course, we know that wind is really an effect of air molecules moving through space. That is, the medium for wind is the atmosphere. This gives us deeper insight into what wind is and how it works. This is the medium perspective.
According to the book, we don't know what the media for the elementary particles are or if there even are any. Our intuition based on waves that we see in everyday life tell us that there must be some medium through which the wave can propagate, but thus far we have found no such medium for waves such as light.
We just know there are measurable properties that we can measure across points in space and we have created mathematical objects (fields) to represent this. From there, we can construct theories and make predictions based on these models.