What does a fundamental particle really look like? After reading a lot and trying to understand people explain it, I made an image in my mind that "fundamental particles are a given position in space to which is given properties", these properties being the spin, mass, energy, etc, and not forgetting that even 'space' itself is formed  by particles. 
Is that description correct, or at least have I come close to understanding it? If it's not right could you please help me see how it works (of course if it is possible to explain)? 
 A: This is an answer by a physicist who worked in expermental particle physics.
Physics is about observing and measuring real numbers which are modeled mathematically in various ways and, very important, the model has to give correct predictions for all new setups of experiments. 
Particle physics is the study of the very small dimensions where quantum mechanics theory holds true , and where, in the models, elementary particles, like the electron, are posited to have zero dimensions and characterized by various quantum numbers . The present day model that fits all the data up to now is the standard model, and the particles are seen in the table.

So the "look like" for a particle physicist starts with the table, i.e. zero mass particles characterized by a four momentum vector when interacting with other particles, the interactions constrained by the standard model.
The "look" is the experimental setup, which measures the interactions and "reports" on the existence or not of the particle. A recent example is the discovery of the Higgs meson, which  discovery validated a long term prediction of the standard model.
If you want to look at the way  particles behave in a bubble chamber , have a look here, where their footprint looks like macroscopic particles would look leaving a trace. The model fits their primary interaction with a quantum mechanical model where there exist probability distribution, which predict a wave like behavior in the probability of scatterings. Detectors and "look" have become more complicated in the effort to see interactions at high energies ( small distances).
The standard model is very successful and it is a quantum field theoretical model. Its calculations are  based on assuming that the whole of space at every point is described by a field, mathematically modelled as a plane wave wavefunction of the elementary particles in the table, on which differential operators operate.The operation is either creating or annihilating particles and is the basis for the Feynman diagrams, i.e. all calculations of the standard model. 
The success of the calculations leads a large number of physicists to believe that the field assigned to a particle is fundamental and "real" . In the mathematical model this is true.  In my opinion, one should not treat mathematics as the underlying reality and posit that it molds nature. One studies nature and finds models that fit the observations. When there is falsification, the mathematical models change, not the data. 
A: Quantum fields are more fundamental than particles, so you should be worrying more about what a quantum field “looks like”. Particles are just field quanta. You can have a field without a particle, but not vice versa.
A classical field can be thought of as one or more numbers at each point in space, which can vary with time. For example, an electric field is a simple vector at each point, pointing in some direction in space.
A quantum field is harder to visualize, because what “lives” at each point is not a set of numbers but an operator on an abstract mathematical space. But the more important thing is that the field extends throughout spacetime.
For example, all the photons in the universe are quanta of the same electromagnetic field. And there just one electron field extending through the universe, which has all electrons as its quanta.
Quantum field theory can be difficult to understand, but it is ontologically simple in the sense that only a handful of quantum fields (17, if you count them in a particular way) can explain a vast amount of what we observe.
By contrast, when you think in particle terms, there are something like $10^{85}$ of them, and that’s just in the part of the universe that we can see!
