They're both types of eigenvectors. Recall that an eigenvector is an element of a vector space V, which is associated to a linear operator A on that vector space (ie, we say it is an eigenvector of that operator), with the property that Av=av, where a is a constant.
An eigenstate is a vector in the Hilbert space of a system, things we usually write like |[itex]\psi[/itex]>. An eigenfunction is an element of the space of functions on some space, which forms a vector space since you can add functions (pointwise) and multiply them by constants. Specifically, you're probably talking about wavefunctions, and operators like x and -ih d/dx.
In the case when the system has no spin degrees of freedom, this wavefunction is just a particular representation of the state, and so eigenfunctions and eigenstates are basically the same thing. So, in most cases you'd be fine not to distinguish them. If there is spin, the full state consists of a wavefunction together with the spin state. Thus it is possible the state is an eigenstate of some operator without the underlying wavefunction being an eigenfunction (indeed, the operator might not even be something that can act on the space of functions, like a spin matrix).
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