Start in the Reminders application and create a new list. To do this, tap the Lists but-ton at the top-left corner of the application. It looks like three lines on top of each other. Then tap Edit > Create New List ... and enter the name Groceries. Click Done. Once you have added a new list, you can refer to that list in Siri.
After creating the list, you can add items to it with simple requests whenever you think of something new you need to buy. Tell Siri "Add eggs to my Groceries list".
Siri asks you to confirm the new item. Just say Yes, and Siri adds it for you.
When you're at the market, just check off the items as you buy them. Couldn't be easier.
One more thing? If you're on a diet, this is an excellent way to keep logging what you eat, a little at a time.