In lattice theories, fermion fields experience (at least) a doubling of the number of particle types in a lattice.

A lattice is a periodic arrangement of vertices. If you Fourier transform a lattice, the space of momenta is a torus with the shape of the fundamental domain of the reciprocal lattice.

This means if we look at the wave solutions over a lattice, the energy (aka frequency) as a function of momentum (aka wave vector) has to be periodic.

For a bosonic field, the action is quadratic and so, the energy tends to have the form

or something like that where m<<1/L. At scales much larger than the lattice spacing (i.e. at low energies) only the momenta around k=0 dominate and we have a single species of boson.

Fermions, on the other hand, are described by first order equations. So, we might have something which goes like

at least with one spatial dimension, but the higher dimensional cases are analogous. If we look at the low energy limit, we see two different regions; one about k=0 and the other about k=π/L. They behave like two different kinds of particles. This is called fermion doubling.

Fermion doubling is a generic consequence of local actions and Hamiltonians. However, the Ginsparg-Wilson action , which is nonlocal solves it (partially, by reducing the number of species).