To mark up in retail by $20$%, do I add $0.20$ times the original cost, or divide by $0.80$? Why is it that when I take a cost of say $\$15.60$ and want to mark the item up at retail 20% that I'm being told two different ways with two different answers? 
The first way (my way) would be to multiply the original cost by $.20$ to get $20$%, then simply add that number to original cost. The second way is to take the original cost and divide it by $.80$ and the number you get ends up being the cost. But this second way is more than the first. The second way almost adds an extra $5$%. 
Why are these different, and which is the proper way to mark up by a percentage?

The second calculation you described is incorrect.
If I say, "Mark up by 20%," that generally means to increase the item's cost by 20% of its original value; i.e., if the item costs 100 dollars, then a 20% increase means to add 20 dollars to its original value, for a total cost of 120 dollars.
To see why the second calculation doesn't make sense, suppose I wanted to double to cost of the item:  this represents a 100 percent markup.  If you tried to do that with your second method, you would end up dividing by zero!
So, what does the second calculation represent?  The result of such a calculation represents the price of an item that, if discounted by 20%, would give you the original price.  So if the original price was 100 dollars, 100/0.8 = 125 dollars.  This is the price of an item whose 20% discount would equal 100 dollars.  And now we see why the previous calculation resulted in division by zero:  if I asked you what price an item would need to be such that a 100% discount would make the new price 100 dollars, clearly that's a nonsensical statement, for a 100% discount means to make the item free.
