Promo Codes and Client discounts are similar in that they don't have a discount scale, so the interfaces to set them up are almost identical. I will describe the Promo Code option in full, expecting that the Client Discount will explain itself.
It starts here:

Click "Discounts" in the left hand sidebar, then "Add Discounts" at the top of the page.

Click "Promo Discount".

Fill out the form:

You can choose to show information about this discount on the widget by checking "info in widget"
NB: The "Description" field will show as a popup, so you can put plenty of information there.
Save the discount, it shows up in the discount list: 

Now I test the Promo Code discount:

By clicking on "Details", they pop up.

Entering the promo code can be done on the last step of the widget, once the code is entered and "Apply" clicked the order changes to something like this:  

Click next then finalizes the order, as usual. 

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