Object where the mouse is

this should be what you want.

    gameObject.transform.position = Camera.main.ScreenToWorldPoint ( Input.mousePosition );
 

Try using Input.mousePosition, it is a vector3 of the mouse position.

You can use a raycast that originates from the camera, whose direction is the mouse's position:

 var hit:RaycastHit;
 var mask:LayerMask;
 var ray = Camera.main.ScreenPointToRay (Input.mousePosition);
       
         if (Physics.Raycast (ray,hit, 500,mask)) {
           //do something at hit.point, which is the point at which the raycast hit a collider
         }
       

EDIT: I apologise, i misunderstood your question. I will keep my answer up however in case it helps.

Both @Professor Snake and @Melanina solutions will work, but what they they wrote will only take you half way. For using ScreenToWorldPoint() you need to set the Z parameter to the distance in front of the camera you want to find. If your camera is looking at positive Z, this code would be attached to the paddle (Untested):

 void Update() {
     Vector3 v3Pos = Input.mousePosition;
     // Get distance the paddle is in front of the camera
     v3Pos.z = Mathf.Abs(transform.position.z - Camera.main.transform.position.z);
     v3Pos = Camera.main.ScreenToWorldPoint (v3Pos);  
     // Don't want to move the paddle up and down, just x 
     v3Pos.y = transform.position.y;
     transform.position = v3Pos;
 }

For @Professor Snake's solution, you would need to construct a mathematical plane at the distance the paddle is from the camera. Then you would use Plane.Raycast() with the ray he constructed to find the screen point in world space.