Jeweler's Stamps
A common stamp that you'll see inside rings are Jeweler's Stamps so you can tell where you bought the ring from. Like "JBR" inside the shank would stand for J.B. Robinson Jewelers, "Pgda" stands for the Piercing Pagoda, Helzberg Diamonds will stamp "Helzberg Diamonds" right inside their rings.
Sometimes these Jewelers will abbreviate their store's names or even use a logo. Every Jeweler is different and uses different markings.
Manufacturer's Stamps
Just like the Jewelers, often you'll see the Manufacturer stamped inside the band of the ring. For example ArtCarved will stamp their rings "ArtCarved" so they know it's their ring and it also helps brand their company, product and image. Take Tiffany for example, they stamp their rings "T & CO".
Manufacturer Stamps can be very useful if you need to find out who made a ring so you could contact them for questions, concerns or even ordering!
Designer
There are a lot of great Jewelry Designer's out there that put their name right inside their rings. A great designer's name can carry a lot of weight. A huge Designer that most people know would be David Yurman which is stamped