In classical algebraic geometry, a generic point of an affine or projective algebraic variety of dimension d is a point such that the field generated by its coordinates has transcendence degree d over the field generated by the coefficients of the equations of the variety.
In scheme theory, the spectrum of an integral domain has a unique generic point, which is the zero ideal.
Oscar Zariski, a colleague of Weil's at São Paulo just after World War II, always insisted that generic points should be unique.
In scheme theory, though, from 1957, generic points returned: this time à la Zariski.
Other local rings have unique generic and special points, but a more complicated spectrum, since they represent general dimensions.