The "Weaver-style base" is a system developed by Weaver for mounting optics and other sighting devices on rifles, shotguns and pistols. The Weaver system is also related to the Picatinny type rails. (Shown is a Weaver-style base for the Buck Mark 22 pistol.)
Several types exists. Commonly the rail is attached with screws into a drilled and tapped receiver. Some rimfire rifles can utilize a tip-off type of conversion rail to allow mounting any optics (like a red dot sight designed for a rail) on a rimfire that only has tip-off grooves.
In general, Picatinny and Weaver rail systems both have slots machined into the rail. One difference often seen between the Picatinny rail and the Weaver rail is the size of these slots. Weaver rails generally have slot that is 0.180" (4.572 mm). The spacing is not always the same, however. Weaver and Picatinny systems differ in other dimensions. Usually you can put Weaver-style rings on either, but some Picatinny optics and accessories may not fit on specialized Weaver-style bases.
This is a great system that make mounting you optics very easy and robust.
For a detailed review of Weaver-style bases and Picatinny rail systems go the the Wikipedia listing by clicking here.
The Tech-Term "Weaver-style" is referred to in the following products: