I read that the first (non-zero) eigenvalue $\lambda_1$ of the Laplacian $\Delta$ on a domain in $\mathbb{R}^n$ or a compact Riemannian manifold $M$ is simple, that is, the corresponding eigenspace is one-dimensional. I believe this depends on the maximum principle and the fact that if there were two eigenfunctions corresponding to $\lambda_1$, then one could produce two nodal domains for each eigenfunction, which would violate Courant's nodal domain theorem. See also the following MO post.
However, on the circle $S^1$, parametrized by $t \in [0, 2\pi]$ with the endpoints identified, $\sin t$ and $\cos t$ are both eigenfunctions for the first eigenvalue, and each has two nodal domains. Is this a one-dimensional phenomenon, or have I misunderstood something regarding the simplicity of the first eigenvalue? Thanks in advance!
I think you've misunderstood the statement you're quoting. What's true is that for a compact Riemannian manifold with or without boundary, the lowest eigenvalue of the Laplacian (with Dirichlet or Neumann boundary conditions if the boundary is nonempty) is simple. Not the lowest nonzero eigenvalue.
In the case of a compact manifold with nonempty boundary, the lowest Dirichlet eigenvalue is positive and simple, while the lowest Neumann eigenvalue is zero and simple (with only the constants as eigenfunctions). For a compact manifold without boundary, the lowest eigenvalue is zero, again with only the constants as eigenfunctions.