What is meant by the term "single particle state"

A single-particle state is a state corresponding to a single particle in isolation. In weakly-interacting translation-invariant systems, for example, a particularly useful set of single-particle states are the plane-wave states $\lvert \mathbf{k}\rangle$, corresponding to a single particle with a plane-wave wavefunction $\langle \mathbf{x} \rvert \mathbf{k}\rangle \propto \mathrm{e}^{\mathrm{i}\mathbf{k}\cdot\mathbf{x}}$. Now, if one considers two distinguishable particles, it is possible for both to occupy the same single-particle state, in which case one would write the two-particle state as $\lvert \mathbf{k}\rangle \lvert \mathbf{k}\rangle$, which really means the tensor product $\lvert \mathbf{k}\rangle \otimes\lvert \mathbf{k}\rangle$. Such tensor products of single-particle states form a complete basis for the two-particle Hilbert space, so that the most general two-particle state (for distinguishable particles) can be written $$\lvert \psi_2\rangle = \sum_{\mathbf{k}_1,\mathbf{k}_2} a(\mathbf{k}_1,\mathbf{k}_2) \lvert \mathbf{k}_1\rangle \lvert \mathbf{k}_2\rangle,$$ where the coefficients $a(\mathbf{k}_1,\mathbf{k}_2) $ are probability amplitudes for finding particle 1 with wavevector $\mathbf{k}_1$ and particle 2 with wavevector $\mathbf{k}_2$. Note that this general form includes all entangled states, where it is not possible to write down a wave function describing particle 1 or 2 alone.

This generalises straightforwardly to $N$-particle systems. A basis for the $N$-particle Hilbert space is given by an $N$-fold tensor product of single-particle states. These single-particle states do not have to be plane waves, they could be whatever you want. When you consider indistinguishable particles, you also have to take into account the bosonic (fermionic) exchange (anti-)symmetry. This is achievable by including only (anti-)symmetrised combinations of single-particle states in the basis set.

Many particle wavefunctions are generally appallingly complicated objects. One way to get a handle on them is to break them down into simpler parts, understand those parts and then put them back together again. We do this by constructing the space of many particle wavefunctions as either a tensor product space or a Fock space.

An obvious way break down a many particle system is to try to consider what each particle is doing individually. Obviously there will be emergent effects in the many body system due to entanglement that were not present when only considering one particle and for strongly interacting systems this breakdown may not be possible, but often it is the only method we have.

So the single particle states are those states which on their own describe a single particle and from which we construct the full space as a tensor product space (i.e. the tensor product of single particle states and linear combinations thereof) or Fock space.