Because Dojo is a toolkit, it's potential uses are unlimited. The only real restrictions are that you're using it in an envornment that uses JavaScript. Though many of you will be reading this in the context of a browser, it's important to remember that JavaScript isn't limited to just that specific use.
With "Ajax" becoming such a popular buzzword, many are looking for an end-all solution to its complexities that not only work well in existing environments, but will allow much more complicated interaction in the future, allowing for very complicated transfer of data between client and server. With the aformentioned widget functionality, and an easy way to implement your own widgets, Dojo works very well alongside HTML.
Dojo also abstracts many of the differences between browsers that have pained developers for years. Things such as differing event objects and HTTP transport systems are a worry that will be forgotten.
Dojo provides a launching point for developing code. This means that you can quickly throw a project together for a client or boss without having to reinvent the wheel. We've built Dojo to be highly reusable, so code you have created for one project can be quickly moved to another without any refactoring.
Responsive applications make many calls to the back-end of web applications which have different call and response characteristics. Standardizing on a portable set of tools for this is beneficial to understanding and predicting overall system behavior.
Because Dojo's widget system sits on top of standard HTML, designers will be able to dive right in with a very shallow learning curve. Dojo allows for designers to add degradable functionality to their (already existing) page without having to implement any logic or strange programming language.
This book describes what Dojo provides and how to use the toolkit. You will find detailed information about the Dojo APIs in the API Reference Doc.