Meshes and Curves

Suppose I have a basic square mesh and I want one side to be curvey. I add a mesh, delete one edge, Add a curve, shape it. But how to I join it to the three sided mesh to make one face?

Blender can only create faces that are either 3 sided (Tris) or 4 sided (Quads). So, in your there is no way to create one face that would have only one edge on three sides and then several edges on the forth side. You will need to fill the shape with quard and tri faces. If you want to post the shape as you have it now and tell if you intend to use subsurf or not, I am sure we can show you several different ways of filling in the shape.