Floquet theory is the study of (quasi)-periodic solutions of the time-dependent Schrödinger equation when the system is subjected to a time-periodic Hamiltonian.$\newcommand{\ket}[1]{\left|#1\right\rangle}$
That is, suppose we have a system under the time-dependent Hamiltonian $\hat H(t)=\hat H(t+T)$. One can show that, because of this symmetry, there must exist solutions of the Schrödinger equation $i \partial_t \ket{\psi(t)} = \hat H(t)\ket{\psi(t)}$ of the form $$ \ket{\Psi_\varepsilon(t)}=e^{-i\varepsilon t}\ket{\Phi(t)}, $$ where $\ket{\Phi(t)}$ is periodic in time and $\varepsilon$ is known as the Floquet quasienergy. In this formalism, one then looks for periodic solutions of the equation $$ \left[i\partial_t - \varepsilon -\hat H(t)\right]\ket{\Phi(t)}=0, $$ which is somewhat easier to solve (as it can be seen as an eigenvalue equation for a differential operator over a bigger domain, instead of a time-dependent Schrödinger equation as such).
Moreover, as a way to analyze this solution, because it is periodic, it is quite common to represent $\ket{\Phi(t)}$ in terms of its Fourier series:
$$
\ket{\Phi(t)} = \sum_{n} e^{-in\omega t} \ket{\Phi_n}.
$$
What is the physical significance of the Fourier coefficients $\ket{\Phi_n }$? Are they solutions of a Schrödinger equation on their own?* Do they have a simple relation to the original Hamiltonian? Do they admit a simple physical picture (e.g. in terms of photon absorption)? What role does e.g. their spatial wavefunction $\left\langle \mathbf r \middle | \Phi_n\right\rangle$ play (or, more generally, the 'direction' of $\ket{\Phi_n}$ in Hilbert space), and what physical significance does it have?
*Upon further reflection, this is almost certainly not the case. Cleaning up §5.5.1 of these notes on Driven Quantum Systems by Peter Hänggi, the Fourier coefficients $\ket{\Phi_n}$ as defined here obey a set of coupled time-independent Schrödinger equations, $$ \sum_{k=-\infty}^\infty \hat{H}_{k}\ket{\Phi_{n+k}} = (\varepsilon +n\omega)\ket{\Phi_n} $$ where the Hamiltonians and couplings, $\hat{H}_n = \frac1T \int_0^T \hat H(t) e^{in\omega t}\mathrm d t$, are the Fourier components of the original Hamiltonian. (Moreover, if $\hat H(t)$ is reasonably harmonic, e.g. with only a sinusoidal dependence on $t$, then only a few couplings with very low $k$ will be nonzero.)
Given this, it seems extremely unlikely that the dynamics of the $\ket{\Phi_n}$ can be reformulated into uncoupled TISEs for each of them, and certainly not in the general case. Furthermore, given the presence of the couplings, it's not clear at all how one can extract a physical significance for the $\ket{\Phi_n}$ from the coupled TISEs.
Also: hat tip to Kevin Tham for pointing out these notes in a comment to this question.
