A central aim of ecology is understanding the mechanisms of community assembly. To address this problem, community assembly is often modeled as a sampling process, in which species are selected from a pool of available species, possibly with effects of interspecific interactions, habitat filtering, and other ecological mechanisms. However, the fundamental stochastic sampling process by which species are selected from the pool remains unexplored. Here we demonstrate the distinctness of four canonical sampling processes, the Bernoulli, Plackett-Luce, multinomial, and fractional multinomial processes. Each process can be affected by ecological mechanisms or it can occur in their absence. Although all four of the processes are a priori plausible and the first two are widely used in ecological models, we show that the multinomial and fractional multinomial processes broadly underlie community assembly.