What are the differences between NoSQL and a traditional RDBMS? What are the differences between NoSQL and a traditional RDBMS?

Over the last few months, NoSQL has been frequently mentioned in the technical news. What are its most significant features relative to a traditional RDBMS? At what level (physical, logical) do the differences occur?

Where are the best places to use NoSQL? Why?
NoSQL stands for &quot;Not only SQL&quot; and usually means that the database is not a relational database, which have been very popular the last decades.
The reason why NoSQL has been so popular the last few years is mainly because, when a relational database grows out of one server, it is no longer that easy to use.  In other words, they don't scale out very well in a distributed system. All of the big sites that you mentioned Google, Yahoo, Facebook and Amazon (I don't know much about Digg) have lots of data and store the data in distributed systems for several reasons. It could be that the data doesn't fit on one server, or there are requirements for high availability.
CAP Theorem
The properties of a distributed system can be described by the CAP Theorem. Of the three properties you can only have at most two:

C​onsistency
A​vailability
tolerance to network P​artitioning

Amazon Dynamo uses Eventual Consistency to come close to get all three properties. The paper Dynamo: Amazon’s Highly Available Key-value Store is worth reading when learning about NoSQL databases and distributed systems. Amazon Dynamo has the A and P properties.
Google take a different approach with BigTable, that has the C and A properties.
Other NoSQL databases
As I wrote in the beginning there are many other kind of NoSQL databases, that are designed for different requirements. E.g. graph databases like Neo4j, document databases like CouchDB and multimodel / object databases like OrientDB.
Finally I would like to say that relational databases will remain popular. They are very flexible and maintainable. But they are not always the best choice.