Dragging a 2D sprite with touch

There are many ways to achieve this, but I recommend the EventSystems DragHandler:

1) Make sure you create an EventSystem component in your hierarchy. 2) Add a Physics 2D Raycaster component to your Main Camera. 3) Write a script that implements IBeginDragHandler, IDragHandler and IEndDragHandler. 4) Add the script to the 2D sprite. Add a 2D Collider as well.

Here is a demo script how it could look. Be aware that the logic in this one might not fit, what you're doing:

 using UnityEngine;
 using UnityEngine.EventSystems;
 
 public class DragAndDrop : MonoBehaviour, IBeginDragHandler, IDragHandler, IEndDragHandler
 {
     public static GameObject DraggedInstance;
 
     Vector3 _startPosition;
     Vector3 _offsetToMouse;
     float _zDistanceToCamera;
 
     #region Interface Implementations
 
     public void OnBeginDrag (PointerEventData eventData)
     {
         DraggedInstance = gameObject;
         _startPosition = transform.position;
         _zDistanceToCamera = Mathf.Abs (_startPosition.z - Camera.main.transform.position.z);
 
         _offsetToMouse = _startPosition - Camera.main.ScreenToWorldPoint (
             new Vector3 (Input.mousePosition.x, Input.mousePosition.y, _zDistanceToCamera)
         );
     }
 
     public void OnDrag (PointerEventData eventData)
     {
         if(Input.touchCount > 1)
             return;
 
         transform.position = Camera.main.ScreenToWorldPoint (
             new Vector3 (Input.mousePosition.x, Input.mousePosition.y, _zDistanceToCamera)
             ) + _offsetToMouse;
     }
 
     public void OnEndDrag (PointerEventData eventData)
     {
         DraggedInstance = null;
         _offsetToMouse = Vector3.zero;
     }
 
     #endregion
 }

There are other ways, like raycasting from the camera into the scene and figuring out what was hit, but honestly I think Unity has a pretty good built-in solution which is much less error prone once you get it working.

Expanding a little on @chrismkhill. The issue with his implementation was if you moved the mouse too fast, you'd lose the sprite

The following would fix that.

     Vector2 mousePosition = Camera.main.ScreenToWorldPoint(Input.mousePosition);
     bool overSprite = GetComponent<SpriteRenderer>().bounds.Contains(mousePosition);
     beingDragged = beingDragged && Input.GetButton("Fire1");
     if (overSprite)
     {
         //If we've pressed down on the mouse (or touched on the iphone)
         if (Input.GetButton("Fire1"))
         {
             beingDragged = true;
         }
     }

     if (beingDragged) {
         //Set the position to the mouse position
         transform.position = new Vector3(Camera.main.ScreenToWorldPoint(Input.mousePosition).x,
                                          Camera.main.ScreenToWorldPoint(Input.mousePosition).y,
                                          0.0f);
     }

Hey there,

If you create a script with this code on a sprite, you should get what you want.

 // Update is called once per frame
 void Update () 
 {
 
 //Gets the world position of the mouse on the screen        
 Vector2 mousePosition = Camera.main.ScreenToWorldPoint( Input.mousePosition );
 
 //Checks whether the mouse is over the sprite
 bool overSprite = this.GetComponent<SpriteRenderer>().bounds.Contains( mousePosition );
     
     //If it's over the sprite
     if (overSprite)
     {
             //If we've pressed down on the mouse (or touched on the iphone)
         if (Input.GetButton("Fire1"))
         {
                     //Set the position to the mouse position
             this.transform.position = new Vector3(Camera.main.ScreenToWorldPoint(Input.mousePosition).x,
                                                 Camera.main.ScreenToWorldPoint(Input.mousePosition).y,
                                                   0.0f);
         }
     }
         
 
         
 }