If I want to form a composite state of, say, an electron and proton's spin states ($\chi_1$ and $\chi_2$, respectively), I can write it as,
$$ \chi_1 \chi_2 $$
This notation hints that we're looking at the product of two functions, and should do our algebra that way. If we were looking at orbital angular momentum eigenstates, then the above could be written as something like,
$$ Y_1(\theta_1,\psi_1)Y_2(\theta_2,\psi_2) $$
That's familiar enough. But! You can't write wavefunctions for spin states. What are these objects, then, and what is their product operation?
