Concept of a point particle in quantum mechanics A point particle is usually thought of as structureless and without dimension. However, given that Heisenberg's uncertainty principle prohibits us from knowing the position of a particle exactly, what is the significance of the concept of a point particle? Are point particles equivalent to elementary particles?
 A: There are two very different issues: the location of a particle in the external world; and location of pieces of the particle inside it (the internal architecture of the particle).
Consider a decisively non-point-like particle, the Hydrogen atom. It has some center-of-mass position $\vec X$ which becomes an operator in quantum mechanics, an observable that obeys the uncertainty relationship with the total momentum $\vec P$ of the Hydrogen atom.
But those overall properties of the Hydrogen atom don't say anything about the internal structure of the atom. And indeed, we know that there are particles inside the atom, the nucleus and the electron. Their distance is comparable to the Bohr radius. To describe this system more accurately, you need to consider the location of both the electron and the proton, $\vec X_e$ and $\vec X_p$. That's equivalent to considering their difference, the relative position $\vec X_{rel}$, and the center-of-mass (weighted average) position $\vec X$.
When you treat the Hydrogen atom as one particle, only $\vec X$ is relevant, the center-of-mass coordinates, and the uncertainty principle relates its error to the error of the total momentum. However, at this level, the uncertainty principle says nothing about the relative position of the electron and the proton. However, one may measure the internal size of the atom – the distance between the electron and the proton – by trying to squeeze many atoms in the same place. In this way, we may find out that the internal size is comparable to the Bohr radius.
It's analogous with other particles although their internal structure may show that they're composed of many  more particles than 2, for example protons and neutrons. Those are composed of 3 valence quarks - but there are also additional gluons and quark-antiquark pairs inside them, too. The protons are also 10,000 times smaller than the atoms.
Elementary particles of the Standard Model - namely leptons (electron, muon, tau, neutrinos), quarks, gauge bosons (photon, gluon, W/Z bosons), the Higgs boson, and also gravitons (outside Standard Model), aside from hypothetical particles in beyond-the-Standard-Model theories - are thought to be point-like at the current level of physics. So you could squeeze an arbitrary number of them into the same place; their mutual interactions seem to obey the uniform power laws up to arbitrarily short distances. That doesn't mean that their center-of-mass position $\vec X$ is well-defined; it only means that the interactions between these particles and others don't dramatically change at some radius that would be analogous to the Bohr radius. You must carefully distinguish claims about $\vec X$ (the center-of-mass location) and $\vec X_{rel}$ (some information about the relative position of pieces inside the article).
At ultrashort distances such as the Planck length, all these particles surely have some structure, e.g. they're composed of a string. But this Planck length is not accessible by direct experiments.
A: An elementary particle is not equivalent to a point particle, if by that you mean a point particle as an idealized classical object in a classical mechanical model. Such a point particle identifies a point in a classical 3-dimensional space as "where the particle is".
The process of converting a classical mechanical model into a quantum mechanical is called quantization, but it isn't "an equivalence". There are any number of ways of talking about what quantization "is", none of which is entirely acceptable (otherwise we wouldn't still be arguing about what it is a hundred years after Bohr's model of the atom), but quantization certainly converts an essentially deterministic classical mechanical model into a quantum mechanical model that is essentially statistical (if no-one comments on the inadequacy of this "certain" statement I will be surprised, but it's vague enough to be almost meaningless).
Although many High Energy Physicists talk about point particles, it has a very specific reference to detailed properties of the mathematics. The most common formal definition of a quantum particle is due to Wigner, for which an idealized point-like event caused by a quantum particle is equally likely to occur at any one place as it is at another. Such a quantum particle might be said loosely to be nowhere in particular, insofar as we might be willing to say loosely that it's everywhere at once. Real events are not point-like (they are regions in detectors that are as small as we can make them, but they are still large numbers of atoms), but we can treat them in mathematical models as point-like, the same way as we can treat the moon as a point particle unless we need more accuracy than such a model allows. In terms of this kind of formal definition of what a particle is, a particle is 3-dimensional, not zero-dimensional; alternatively, we can say that such a particle is point-like in Fourier space. For an algebraically minded person, point-like in Fourier space is as good as point-like in real space, but it's an image that should be worked through in terms of the algebra, it's not the same as an ordinarily imagined point in space.
The problem with the Wigner definition is that it only works for free fields, for which there are no self-interactions or interactions with other fields. High Energy Physics works with what are called asymptotic fields, which are interacting fields "with the interaction turned off" a long time before and after an interaction, so that in HEP we can only say that we measure the asymptotic fields/particles. The asymptotic fields/particles can be said, in the roundabout way I've tried to describe above, to be point-like, but the interacting fields cannot.
There are currents in Physics towards saying that there are only (quantum) fields, although explicit statements to that effect tend either to be off-hand or not adequately detailed. A collection of such statements can be found at the beginning of a recent article by Art Hobson, http://arxiv.org/abs/1204.4616, "There are no particles, there are only fields", which makes a worthy attempt to cash out the idea.
A: A point particle is the idealization of a real particle seen from so far away that scattering of other particles is as if the given particle were a point.
Specifically, a relativistic charged particle is considered a point particle at the energies of interest if its interaction with an external electromagnetic field can be accurately described by the Dirac equation.
The deviations from pointlikeness are usually described by means of 
form factors that would be constant for a point particle but become 
momentum-dependent for particles in general. For example, the electric form factor is (essentially, in the nonrelativistic case) the Fourier transform of the electric charge distribution in space with the same scattering behavior as observed for the given particle. It would be identically 1 for a point particle. 
The form factors contain everything that can be observed 
about single particles in an electromagnetic field.
In particular, the charge radius is defined as the number $r$ such that 
the electric form factor has an expansion of the form 
    $F_1(q^2) = 1-(r^2/6) q^2 $  if  $r^2q^2\ll1$.
(Units are such that $c=1$ and $\hbar=1$.) This definition is motivated 
by the fact that the average over $e^{i q \cdot x}$ over a spherical shell 
of radius $r$ has this asymptotic behavior.
Due to radiative corrections form the renormalization procedure, elementary particles (as defined in QED or the standard model) are not quite point particles (one says that they are point-like); for example, the charge radius of the electron is positive according to calcuylations by Weinberg.
(See Section 11.3 of: The quantum theory of fields, Vol. I, 1995.)
The relations between form factors for spin 1/2 particles and 
terms in a modified Dirac equation describing the covariant dynamics
in an electromagnetic field of a particle deviating from a point 
particle are given in 
   L. L. Foldy
   The Electromagnetic Properties of Dirac Particles
   Phys. Rev. 87 (1952), 688 - 693.
For more details, see Section ''Are electrons pointlike/structureless?'' from Chapter  B2: Photons and Electrons of my theoretical physics FAQ
