
In most cases the Attributes Toolbar is the primary control. These snapping actions are 'soft' as the graphic may be 'pulled' away from the snap point, interactively, by continuing the mouse dragging action. You may want to read up on that in the help, and possibly turn it off. The graphic or vertex that is in motion 'jumps' to the precise position when the the graphic is close to the snapping reference point. On the right the red vertex is outside the snapping tolerance (despite being closer to the edge). On the left, the red vertex is within the snapping tolerance to the blue vertex, so it will snap. the small square at the junction of the Gizmo axes. The function snaps to the vertices of geometries only. From your screenshot It looks like you have "Move in screen space" turned on (Customise > Preferences > Gizmos (tab). then press and hold the c-key and the move manipulator will change to a circle at the center to indicate snapping is on. so to snap a set of verts to a curve you'd. In addition to a great deal of other good stuff, there is a hefty section devoted just to this, and associated, topic(s). press and hold the following keys to select the target of the snap. If you get totally stuck with this and can afford $99 - get the 3DBuzz "Max Fundamentals" DVDs. Remember that it is the cursor which is snapping, not the object's pivot (unless you grab the gizmo in which case they are the same point). Another possible solution (but not 100 as option 1) is to. Then from there all other objects should vertex snap to 'Object A'.

You can see it quite clearly by watching the cursor as it snaps to other vertices - it maintains the original offset (which happens to be 0 if you grab the gizmo). The best solution is to Open Edit/Snap Settings window. If you grab anywhere else on the object you are introducing an offset. If you grab the Gizmo you are moving the object by its pivot - no offset. Unfortunately, it wasn't clear in your original post that you were moving the object rather than being at Vertex SubObject level and moving the vertex.
