Self-assembly of proteins into amyloid fibrils plays a key role in a multitude of human disorders that range from Alzheimer's disease to type II diabetes. Compact oligomeric species, observed early during amyloid formation, are reported as the molecular entities responsible for the toxic effects of amyloid self-assembly. However, the relation between early-stage oligomeric aggregates and late-stage rigid fibrils, which are the hallmark structure of amyloid plaques, has remained unclear. We show that these different structures occupy well-defined regions in a peculiar phase diagram. Lysozyme amyloid oligomers and their curvilinear fibrils only form after they cross a salt and protein concentration-dependent threshold. We also determine a boundary for the onset of amyloid oligomer precipitation. The oligomeric aggregates are structurally distinct from rigid fibrils and are metastable against nucleation and growth of rigid fibrils. These experimentally determined boundaries match well with colloidal model predictions that account for salt-modulated charge repulsion. The model also incorporates the metastable and kinetic character of oligomer phases. Similarities and differences of amyloid oligomer assembly to metastable liquid-liquid phase separation of proteins and to surfactant aggregation are discussed.