To start a Smart Group, go to File > New Smart Group. A dialog will drop down in Address Book for you to start entering criteria for filtering. For example, you could create a group of everyone whose birthday you have in your contacts by choosing the item Birthday from the first drop-down menu item, then adding the menu item "is set." Note that you can't create a list of people with birthdays in June, which is a bummer. But the Smart Groups have a variety of ways to filter, some more useful than others and several are dependent on the data (dates vs. text, for example).
To delete a group you'll have to make a trip to the menus, as no amount of right-clicking or key pressing will do it. Delete is in the Edit menu, under Delete Group.
One more thing: the notes field in Address Book extends the power of Smart Groups just a bit. After returning from WWDC I added the business cards of people I met and added the note "wwdc" to each one. I then made a Smart Group where the Note contains 'wwdc' and now I've got an easy way to see the group of people I met at WWDC. Think of the Notes field as a loose tag field, if you like.
I've also set up my Address Book to add family members based on a list of surnames, but that may only work if you have an oddball last name like mine!