A state diagram is a type of diagram used in computer science and related fields to describe the behavior of systems. State diagrams require that the system described is composed of a finite number of states; sometimes, this is indeed the case, while at other times this is a reasonable abstraction. There are many forms of state diagrams, which differ slightly and have different semantics.
Stateful means the computer or program keeps track of the state of interaction, usually by setting values in a storage field designated for that purpose. Stateless means there is no record of previous interactions and each interaction request has to be handled based entirely on information that comes with it.