I've read that the Jordan-Wigner transformation changes qubits into spinless fermions. What, exactly, are spinless fermions? I'm guessing it doesn't mean spin zero which would be a boson, so what does it mean?
It is only in relativistic quantum field theory that spin and statistics are connected; see http://en.wikipedia.org/wiki/Spin-statistics_theorem
In non-relativistic QFT the two are completely distinct. Spin manifests as local degrees of freedom, and statistics is encoded in the (anti-)commutation relations at equal times.