In the discussion about fermions, the concept of a wave function appeared, to explain what makes fermions so very strange under a change of coordinate systems. The analogy of particles with waves and oceans has been made also already quite a bit back. It is about time to be just a bit more precise about what a wave function and a field is for a theoretical physicists.
Go back to the idea that particles emerge a some waves at a particular point on an ocean. Two particles would then be just two such waves at two different points. Now the underlying concept appears just to be the ocean, rather than the waves. And indeed, the waves can very well be identical.
That is the underlying idea also in theoretical physics - not only particle physics, but this permeates many ares of theoretical physics: The basic object is the ocean. In the context of particle physics, this ocean is then called a field. Such a field is now existing at every point in space and at every instance in time. In the very literally meaning of the word, it fills up all of the universe. If there is nothing of interest around, this is because the size of the field at this point in space and time is small or even vanishing. However, if there is a spike at some point in the field then just as in the picture of the ocean there sits a particle. If there is a second spike somewhere else, then there is another particle, and so on. Since all the spikes belong to the same field, they describe the same type of particle, say an electron. The spikes may move with different speeds, so the electrons appear to have different speeds, but they are still electrons. That is the reason why all electrons are the same: They are just spikes in the same field. Such a spike is often called an excitation of the field, and this excitation is the electron.
Then what is about the other types of particles? The quarks, the gluons, the Higgs? Well, these belong just to other fields. That is, our universe is filled up with many fields, all existing simultaneously at every point in space and time.
You may be wondering how this should work, and if this is not a bit crowded. But you know already that fields are mathematical concepts. For example, you can associate with every point in space and time a temperature, and thus create a temperature field. At the same time, there is an atmospheric pressure field. Both can happily exist simultaneously. But they are not ignoring each other. As you know, both a related with each other: If either changes this indicates a change of the other as well. Though this analogy is not exactly the same as the particle physics fields, and there are more things involved, the basic idea is the same.
Also the particle physics fields interact, and thus not ignore each other. Their interaction can be more or less translated once more from the analogy with the waves, which has been discussed earlier. So, in this way, everything is realized we see in particle physics. There are fields for every type of particle, which may interact. We are then 'just' a very complicated, combined, and correlated simultaneous excitation of all of these fields, as is your desk or your computer.
Now, what are the wave-functions? Well, in the beginning, quantum physics was formulated not taking into account the effect of large speeds, i.e. of special relativity, something I will explain in more detail later. In this case, the concept of fields can be reduced to instead describing only the waves making up a single particle. In principle, you isolate each wave describing a particle, and discuss it alone. These mathematical quantities describing these single particles are then called wave functions. So wave functions can be thought of as the slow-speed limit of the fields, when all particles are treated separately. Mathematically, this is not quite precise, but should give a rough idea.
Now it is possible to come back to fermions. When you rotate the coordinate system once, it is this wave function (or the field), which change not directly back to the original, but only after a second rotation. Of course, nothing you can actually measure (or experience) changes when rotating your coordinate system once fully. That is because the wave function or the fields cannot be directly measured, just things we can derive from them. However, the underlying fact that you have this obscure change influences the properties of fermions, and leads, e.g., to the Pauli exclusion principle.