Storytelling is a key element in any artistic medium. Good storytelling can drive your product (be it an artistic creation, an actual story, a film, a physical product, etc) to success. The same can be said when creating a meaningful brand for a company or product. In order to catch an audience's attention, and most importantly--keep it, you need to establish parameters that will allow the audience to connect on a personal level. If the audience isn't made to care (bad writing in a movie, for example), your product will not fair well. This is where good storytelling comes in. Depending on the product, you don't necessarily have to write a dramatic ad campaign a la Don Draper (although I'm sure it wouldn't hurt if you have the time and resources), but you do need to create strong visual storytelling to cement a brand's image.
Take iperdesign's recent brand repositioning of CTI as an example. The colors blue, red, and green are repeated throughout CTI's website to solidify the meaning of each: blue for sourcing, red for management, and green for technology. This visual line of thought is consistent throughout CTI's website to present a continuity of strong brand image.
Some brands or products may inherently lend themselves to the goal of creating a connection with the audience. Iperdesign's Cupido app, for example, connects to its audience on a personal level. The story that is told with this app and its branding materials is that the user will not only discover something about herself, but the user will also discover something about her relationships with others. The very nature of the app allows for a personal connection with the user--there are stakes that the user cares about that are presented in a clear manner.
One of the main components to keep in mind when storytelling through branding is consistency. When an author writes a novel, she doesn't suddenly change the plot or characters half way through. The same idea applies to branding within a new marketing campaign or product release. Consistency not only helps the audience to easily identify and remember a brand, but it also aids in communicating what that brand means to the audience. A person viewing the CTI website will see blue and immediately know what it means because of the repeated association. This idea is visual storytelling, in a way.
Consistency in your "story" can lead to branding success; you'll have a happy client, and a satisfied user who cares about the content.