The state of an application (or anything else, really) is its condition or quality of being at a given moment in time—its state of being. Whether something is stateful or stateless depends on how long the state of interaction with it is being recorded and how that information needs to be stored.
Stateless
A stateless process or applicaton can be understood in isolation. There is no stored information or references to past transactions. Each transaction is made as if from scratch for the first time. Think of stateless transactions as a vending machine: a single request and a response.
Stateful
Stateful applications and processes, however, are those that can be returned to again and again, like online banking or email. They’re performed with the context of previous transactions and the current transaction may be affected by what happened during previous transactions. For these reasons, stateful apps use the same servers each time they process a request from a user.