Here is a notification from Snapchat explaining what a Snapchat My Story is:
Snapchat stories have created much more of an interaction between Snapchat friends because you are able to send your story items to all of your Snapchat friends in one click of a button. Snapchat stories have also created new ventures for Snapchat.

Snapchat's newest "money makers" are sponsored Snapchat stories. The first sponsored story was a collaboration with Samsung at the American Music Awards. This story included photos and videos of celebrities, the show, and behind the scenes of the show with a sponsored illustration of the AMA's and the Samsung Galaxy Logo. I personally think this is a genius advertising strategy because you are not throwing it in the users face. For example, if they don't want to watch the story, they simply don't. Here are some screen shots from the first sponsored Snapchat story.
A great strategy that Snapchat has is that not all of the Our Stories are sponsored events. For example, no company is paying them to create a story about National Hot Air Balloon Day, but it creates an interest from the audience. I think this is a great new way to advertise to users through Snapchat.
Source: http://techcrunch.com/2014/11/23/snapchat-sponsored-our-story/