Several ways. The most common is extrude: select some vertices, type e to extrude, (select region, edges or vertices only) then move the new vertices to where ever you want them. These new verts will be connected by edges or faces to the originally selected verts.
Select an edge or a face, and subdivide (type w, select subdivide) to get new vertices on the edge or face. This may create triangles in surrounding faces as it adds new verts.
Select some vertices and Shift d to duplicate them. Move them to where ever you want them, and connect them up using the make face tool (type f).