What does the notation $^1 S_0$ represent in particle physics?

These are term symbols, which are a compact way to encode the angular-momentum characteristics of a quantum state.

The general scheme for this notation is of the form ${}^{2S+1}L_J$, where:

• $2S+1$ is the spin multiplicity, i.e., the number of linearly-independent spin states of the system.
• $L$ is the total orbital angular momentum of the system, denoted in spectroscopic notation (so the values $L=0,1,2,3,4,5,ldots$ are notated $S,P,D,F,G,H,ldots$).
• $J$ is the total angular momentum of the system, obtained by combining the total spin and total orbital angular momentum quantum numbers, $S$ and $J$, using the quantum-mechanical procedure for the addition of angular momenta.

The fact that the quantum numbers are upper-case ($S,L,J$, as well as the upper-case letters $S,P,D,ldots$ in the spectroscopic notation) indicates that they denote the total values of a multi-particle state. For single-particle quantities the usual convention is to use lower-case letters and symbols.

It's probably fair to say that term symbols are most commonly used in atomic physics, and there they represent the various angular momenta of the electrons in the system. However, the notation is generic, and it can also be applied to particle-physics contexts, and there they could be used to denote some specific scattering state, as well as, say, the way in which the angular momenta of a set of quarks combine to make a bound state (that we then call a particle). The specific details of exactly which angular momenta are being referred to will, of course, depend on the context where the term symbol appears.