In plastic surgery, a lot of time can be spent in the preop area marking your patient. So you want a marker that won't wash off so easily that it is gone with the scrub. For breast and body "work", I use (and the nurses tell me so do most of the others) a black Sharpie.
It is marketed as a permanent marker, but I still find that I have to remind the person prepping the patient to not "scrub too hard" or "that's enough". When you have marked the patient standing or bending in different ways to be sure you get the most skin removed, these positions and maneuvers can't be duplicated in the operating room. During the procedure, I use whatever marker the hospital has, usually the Accu-line products. Those are also what I use when I need a really fine line (ie eyelid) when marking.
The skin marker should be nontoxic and non-allergenic. If used during the procedure, then it must be sterilizable. The ink must have a visible color and must be non-reactant with other chemicals used on the skin (e.g., povidone iodine). The ink must be resistant to mechanical cleaning but removable in time.
In quilting, you want a marker that will stay long enough to see the pattern you are quilting. You want to be able to either "brush" it off gently later (as with chalk pencils) or to wash it out. The different colors of fabrics used can sometimes make this more challenging. For the quilt I am preparing for hand quilting, I used the blue washable marker on the "mustard" (light fabric) and a silver chalk pen on the brown (dark fabric). Here are some links to tips by experts like Ami Simms (blue washable marker), and Sharon Darling (Quilter's Choice Marking Pencil, Miracle Chalk).
Skin Marking in Plastic Surgery; Plastic & Reconstructive Surgery, 115(5):1450-1451, April 15, 2005; Ayhan, Meltem M.D.; Silistreli, Ozlem M.D.; Aytug, Zeynep M.D.; Gorgu, Metin M.D.; Yakut, Macide M.D.
Quilt Tips From Quilters Around The World--Marking Tips
Appalachian Mountain Quilters Marking Techniques by Kimberly Wulfert