How to make object not be move into another object

I have a game in which the player can use their mouse and move objects around with it. The problem is, if they move the object down, instead of sliding downwards on the plane, it just goes straight through it instead.

This is a problem, since the z coordinate of an object being moved cannot be easily manipulated, since instead of sliding downwards and decreased the z coordinate, it goes straight through and pops back up again.

I want to make a sort of clamp so the object being moved cannot be moved inside the plane, but instead can only be moved on it.