With Apple Pay, you can save your payment information in a digital wallet, which you can use to pay for things like a regular debit card. It's most useful on cash registers that support it, where you can pull the phone instead of your wallet, but it also works on the Mac when you shop online.
Apple Pay works seamlessly on newer Touch Bar Macs, but if you have an older Mac, you need an iPhone or Apple Watch for authentication of Touch ID. Apple Pay currently only works in Safari, but it may change in the future.
Setting up Apple Pay on your Mac
The first thing you have to do is activate Apple Pay in Safari. Open Safari settings, switch to the "Privacy" tab and enable "Let sites check if Apple Pay is set up".
You should have been asked to set up Apple Pay after you first install your Mac, but if you have not added your card then or if you need to change something, you can do it when anytime, open the system settings and click on the "Wallet and Apple Pay" option. 
From here you can add new cards, remove cards, change your default card, change your billing and shipping address and see your latest transactions.
To add a new card, just click the + button at the bottom:
It will try to automatically read your card information from the camera, which is a good feature, but if it does not work can you manually enter it?  If you have an older Mac
If you do not have a Touch Bar Mac, you can not configure Apple Pay if you do not have an iPhone. You must add your card to your phone and then enable "Allow payments on Mac" from the Wallet and Apple Pay settings on your phone.
You will be able to make purchases with Apple Pay from your Mac, but you must pull out your phone or watch to verify with Touch ID. A bit less seamless, but at least it is compatible.
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