For a free particle, there is a corresponding free wavefunction freely developing through spacetime. A wavefunction can be broken up in Hilbert space just as a vector in ordinary space can be broken up into its components. The base vectors of the Hilbert space are the eigenfunctions (eigenstates) of an operator acting on the wavefunction.
When a measurement is made the wavefunction is projected onto one of the eigenstates (with the corresponding eigenvalue, the result of the measurement). When we make a position measurement, though, the wavefunction ends up being localized in a small volume. There is not just one value of position, but a continuous small range.
It is, in theory, possible to make a precise measurement, giving us a precise value of the position. The corresponding "eigenfunctions" are said to be Dirac delta functions. Or, better, Dirac delta distributions.
Now I've read many times that these "eigenfunctions" (Dirac deltas) are not really quantum mechanical states. For example, @ACuriousMind writes in a comment to this question:
It might be worth it to add a caveat about |x⟩ and |p⟩ not actually being states
While @LubosMotl writes:
Yes, they [the Dirac deltas] are the (not normalizable) eigenstates of the operators. And some things are not stated somewhere because not every book lists all true statements.
I'm not sure if these two comments contradict each other. An eigenstate has to be normalizable. Motl writes that this is not the case. Is this what ACuriousMind means when he says that eigenstates of position and momentum don't exist? Is the fact that the Dirac delta is a distribution (and not a function) of importance?
Does this mean that the wavefunction can't be written as a superposition of states (regarding the position or momentum operator)? Somehow infinity is involved, I guess.
It could also be that it's just a matter of words. A particle has to be in some "state".