Suppose we have a continuous medium, an elastic band or the surface of a pond, we can then have traveling waves propagating in the medium. Just throw a stone into a calm pond and observe those waves. As long as the equations governing the medium are linear, we can superpose solutions and create pulses: localized bundles of energy which look a lot like particles, almost. So, it looks like we can throw away the problematic point particle, replacing it by a special kind of field, the one that obeys linear equations. Those are called linear fields, or free fields. However, we are not yet there…
Two things seem at first to distinguish particles from wavepackets, the ones you form from free fields. First, the energy in a wavepacket can be anything at all, whereas particles come in discrete units. You can have one electron, two electrons, but not 2.3 electrons. Secondly, electrons are microscopic: they are tiny, almost pointlike, and individual electrons carry minuscule amount of energy. Waves are usually thought of as macroscopic phenomena involving lots of moving parts.
Both these issues call for quantization. When quantizing the waves of a free field we get a discrete, equally spaced spectrum. The problem is actually identical mathematically to the familiar harmonic oscillator. We are then free to call those excitations of the field particles: they carry small amounts of energy, and that energy comes in multiples of some basic quantity, which is then naturally interpreted as the energy of a single particle. This is the miracle of particle-wave duality: quantization of waves in a linear field give rise to particles. We don’t really have to have particles as separate from fields, all we need is fields.
Suppose now the equations governing the motion of the fields are not exactly linear. If they are nearly linear, we will get nearly the same picture as before: we will have excitations looking like one particles, 2 particles, 3 particles…but the energy is not going to be precisely equally spaced. In that case we can just say the particles are not precisely free: when putting a few of them together the total energy is slightly different than the sum of the individual energies. This is because they feel each other’s presence, they interact. As long as the energy associated with that interaction is small compared to that of individual particles, we have a recognizable picture of small particles moving about and interacting.
But, as before, fields are much more complicated and interesting than particles. In some circumstances, when their equations are linear, or nearly so, they can give rise to particles. In other circumstances they do not: think about the pond on a windy day, or on a cold January day in Ontario… If the fields are strongly interacting, or are under unusual circumstances, they exhibit a rich variety of phenomena, none of which could be interpreted in terms of particles. The study of QFT is the study of those phenomena.