NoSQL encompasses a wide variety of different database technologies that were developed in response to the demands presented in building modern applications:
Developers are working with applications that create massive volumes of new, rapidly changing data types — structured, semi-structured, unstructured and polymorphic data.
Long gone is the twelve-to-eighteen month waterfall development cycle. Now small teams work in agile sprints, iterating quickly and pushing code every week or two, some even multiple times every day.
Applications that once served a finite audience are now delivered as services that must be always-on, accessible from many different devices and scaled globally to millions of users.
Organizations are now turning to scale-out architectures using open source software, commodity servers and cloud computing instead of large monolithic servers and storage infrastructure.
Relational databases enforces ACID. So, you will have schema based transaction oriented data stores. It's proven and suitable for 99% of the real world applications. You can practically do anything with relational databases.
But, there are limitations on speed and scaling when it comes to massive high availability data stores. For example, Google and Amazon have terabytes of data stored in big data centers. Querying and inserting is not performant in these scenarios because of the blocking/schema/transaction nature of the RDBMs. That's the reason they have implemented their own databases (actually, key-value stores) for massive performance gain and scalability.
NoSQL databases have been around for a long time - just the term is new. Some examples are graph, object, column, XML and document databases.