What really is a particle? In Classical Mechanics we consider particles as things whose internal structure for the purpose of studying some phenomenon might be neglected. In that setting we associate particles to points and sometimes we picture then as very tiny balls.
In that understanding of particles, they obviously have a well defined position. Also it is a quite easy idea to grasp intuitivelly, after all we see macroscopic things at particular locations.
On the other hand, when we consider the mathematical model of Quantum Mechanics things change a little. To describe a particle instead of giving a location we give a probability distribution which tells probabilities of detecting the particle somewhere.
In that new setting I've found two ways to look at it:


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*The particle is still as in Classical Mechanics: something we can consider as a point and visualize as a tiny ball. In that way, for some reason I don't know the theory don't allow us to associate it with a particular location.

*The idea of particle must be revised, it is not some tiny ball we treat as a point, but rather something spread over a region. In that setting we have to revise what we mean by visualizing a particle to make the statistical interpretation of the wave function make sense.
So which point of view is correct? Considering Quantum Mechanics what really is a particle? And how to bridge the gap between the idea of particle from Classical Mechanics and Quantum Mechanics?
 A: There are no such things as particles in the physical world. The correct description of "small things" in classical mechanics is that the dynamics of the motion of the center of mass of an extended object is the only relevant physical quantity while internal degrees of freedom like rotation, vibration, magnetization, temperature etc.. can be ignored. That leaves us with an abstract triplet of numbers that lack any description of the actual physical size of the object (it could be a planet or a star). QM simply says that this triplet follows a different set of rules, but no particles are required, either. 
Now, we could be having a discussion about why our high school educators fail to present students with a consistent and physically correct notion of the world and why almost everybody seems to be leaving high school with the belief that a plane is a magical flying collection of infinitesimally small balls rather than an extended physical body that has three degrees of freedom for translational movement, three degrees of freedom of rotation and a number of non-trivial internal degrees of freedom for the movement of control surfaces and the sloshing fuel. After all, that is how aerospace engineers are looking at real planes, they for sure are not playing with tiny balls. Neither are physicists when they do physics, by the way. Physicists are talking about the movement of the center of mass in classical mechanics and about quanta resulting from measurements in quantum mechanics. No balls of any sorts at any time!
A: Let's be clear on what we call a particle. 
It is an object of which you can measure its physical properties like energy, momentum, charge or spin. None of those is space and for a good reason : space (or time) is not an intrinsic property of a particle.
Space is a useful mean to describe the universe and the particles in it, nobody could deny that. But as far as anyone can tell space lies in our mental image/theory of the world and cannot be measured in any way. For instance, time is defined nowadays with respect to the frequency of the wave emitted by a spin transition between two energy levels, i.e. It is a measure of energy due to $E=h\nu $.
More importantly, space goes a similar way as it is itself defined through the behavior of light through time. 
This existence of time is a philosophical debate though, and another point of view about it is absoluteness. Newton would have said that time is not human dependent and exists absolutely, whereas Kant or Leibniz would have shared my previous opinion.
Back on track, as there is no way to measure the "space of a particle", it is irrelevant to wonder what a particle looks like in space. A particle simply is a set of observables.
As for the matter of which representation of particles is the best between classical "balls" and quantum waves of probability, the latter would make more sense because a ball would have a border and the definition of such a discontinuity between existence and void would lead to problematic consequences.
Moreover, the representation as a classical ball comes from the large difference in size between the measuring tool and the particle (look up for decoherence). When devising more precise experiments, like Young's slits experiments, the wave-like behavior is exhibited.
To conclude, the best representation of what a particle is depends on what you'll do with it. If you look at a huge set of particles (like studying the motion of a plane), using QM representation is useless. If you look at a small set of particles like biological proteins using quantum representation is a good idea. 
A: The idea that the particle is "spread out" over a region is closer to true.  As you say, in QM the position of a particle is described with a probability distribution.  Only that's not entirely a correct description.  It would be more correct to say that particles don't have position, they have superposition.  The superposition is described using a complex (meaning it uses imaginary numbers) "wave function", the real part of which is the probability distribution you're thinking about.
Trying to think of the particle as "actually" being somewhere is specious, and will get you in trouble.  Likewise trying to think of it as having an "actual" speed is wrong, too.  Instead you have to play with the math of the wave functions.  The math isn't too hard but it certainly doesn't appeal to our monkey brains' intuition!
The MIT Open courseware has a lecture series on QM.  Just watching the first few lectures is immensely useful for getting an idea of how QM works and its quirks.  As I said, the math isn't too hard.  It's more a matter of wrapping your head around ideas that are quite outside the macroscopic world we're built to understand.
A: The way I like to understand particles is with an analogy to a our solar system.  Particles like the stars and planets make up the whole, the whole being something we perceive as a complete entity or system. The whole is the "wave function" and the particles are the units that make up the wave function.
Also, a massless object still contains particles.  To date there isn't a particle-less particle.  Also, I consider space to be the particle-less entity that contains all particles. 
