Resource manipulation operations through HTTP requests should always be considered atomic. All modifications of a resource should be carried out within an HTTP request in an isolated manner. After the request execution, the resource is left in a final state; this implicitly means that partial resource updates are not supported. You should always send the complete state of the resource.
Back to our catalog example, updating the price field of a given item would mean making a PUT request with a complete document (JSON or XML) that contains the entire data, including the updated price field. Posting only the updated price is not stateless, as it implies that the application is aware that the resource has a price field, that is, it knows its state.
Another requirement for your RESTful application to be stateless is that once the service gets deployed on a production environment, it is likely that incoming requests are served by a load balancer, ensuring scalability and high availability. Once exposed via a load balancer, the idea of keeping your application state at server side gets compromised. This doesn't mean that you are not allowed to keep the state of your application. It just means that you should keep it in a RESTful way. For example, keep a part of the state within the URI, or use HTTP headers to provide additional state-related data
The statelessness of your RESTful API isolates the caller against changes at the server side. Thus, the caller is not expected to communicate with the same server in consecutive requests. This allows easy application of changes within the server infrastructure, such as adding or removing nodes.
Remember that it is your responsibility to keep your RESTful APIs stateless, as the consumers of the APIs would expect them to be.
Now that you know that REST is around 18 years old, a sensible question would be, "Why has it become so popular just quite recently?" Well, we the developers usually reject simple, straightforward approaches and, most of the time, prefer spending more time on turning already-complex solutions into even more complex and sophisticated ones.
Take classical SOAP web services, for example. Their various WS-* specifications are so many, and sometimes so loosely defined, that in order to make different solutions from different vendors interoperable, a separate specification, WS-Basic Profile, has been introduced. It defines extra interoperability rules in order to ensure that all WS-* specifications in SOAP-based web services can work together.
When it comes to transporting binary data with classical web services over HTTP, things get even more complex, as SOAP-based web services provide different ways of transporting binary data. Each way is defined in other sets of specifications, such as SOAP with Attachment References (SwaRef) and Message Transmission Optimization Mechanism (MTOM). All this complexity was caused mainly because the initial idea of the web service was to execute business logic remotely, not to transport large amounts of data.
The real world has shown us that, when it comes to data transfer, things should not be that complex. This is where REST fits into the big picture—by introducing the concept of resources and a standard means for manipulating them.