On each object you will choose one or more faces that will be removed. In between these faces will be your connection.
- Select both objects in object mode. Press
Ctrl+Jto join the objects into one. Then enter edit mode and change to face manipulation mode.
- Remove the faces that will be joined. Select them and press
X, remember to delete faces, not vertices.
- Switch back to Vertex manipulation mode. Join the objects by selecting the corresponding vertices from each object and using
Fto create a new face. For example, in the image below I've selected two of the vertices on the sphere and one on the cuboid. When I pressFBlender will create a new triangle face there. It also supports quads if the vertices share a common plane. I typically limit the vertex selection to three or four vertices at a time, I tend to think I know where I want the faces to be better than Blender. But you can try selecting all the vertices and see what Blender does for you.
- Continue the previous step until all faces are joined.
