If you wish to move a layer mask from one layer to another (in other words, remove it from the original layer and add it to another existing one), then simply click on the layer mask and drag it to the layer that you want it to be on. 

(A layer mask is shown next to the layer itself in the layers window.)

move a layer mask
click and drag to move a layer mask

If you want to copy a layer mask from one layer to another, you need to hold Alt [Win] / Option [Mac] + click and drag.

hold Alt to copy a layer mask in photoshop
hold Alt to copy a layer mask

Related Guides:
How to Correctly Crop a Layer in Photoshop
How to Flip a Layer in Photoshop

Video Tutorial to Move and Copy a Layer Mask

For more information on the basics of working with layer masks, feel free to check out this video by First Shot School about copying a layer mask in Photoshop.

Copying a Layer Mask in Photoshop
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