When I first saw this question I thought you were kidding, but when I recently went to delete a bad entry from the Contacts database on my Mac OS system, I realized you were exactly right and that it is a bit tricky to figure out. My guess is that it's because as is often the case at Apple, the designer errs on the side of a streamlined interface rather than one that makes access to the key features overt and obvious.

Fortunately, I did figure it out and it turns out that there are two ways to delete entries from your Contacts [Address Book] application, and if you've got it synced via iCloud, it'll then magically disseminate the update to all of your iOS and other Apple devices too.

To start out, I'll search for the person I want to delete: Ryan Smith-Jones...

You can see I have a fair number of matches for "Ryan" in my Contacts. Turns out that one shortcut, though it's certainly not obvious, is to press delete on your keyboard with the card you want to remove selected. Ya just gotta know it's an option.

More logically, if you Cmd-Click on the person's name, options appear, but none that offer that delete functionality:

This would be a perfect place for "Delete Card" as an option, but instead I'll make the logical choice of "Edit Card". Now it shows up with the ability to change or update the information:

But still, no "Delete" option. Weird.

Click "Done" to stop the edit and instead learn about option #2: The "Edit" menu:

If you guess that "Delete Card" works, well, you're right. But it seems counterintuitive for me to need the menu option when just about every other feature and capability in Contacts is easily accessible from within the app's floating window.

Still, click on it to delete the contact in question and you'll get a confirmation window:

Click on "Delete" and Ryan's gone into the ether, never to return in my Contacts database.

As I said, once you know the trick, it's pretty easy, but I'm still surprised that the option to delete an entry doesn't show up in the context menu or as a button when editing a Card.