at the most basic, "conversational state" refers to the value of instance fields.
For stateless session beans, the container does not guarantee that subsequent method invocations will use the same EJB instance (from the pool), hence you cannot assume that the values you placed when you call a bean method, will still be there when you call the method again (or another method of the bean).
For stateful session beans, the container guarantees that subsequent calls will use the same EJB instance, hence you can keep instance field values.
For the sake of an example, say you have a bean that has an increment() and a retrieve() method. Increment increases the stored value, and retrieve gets the current stored value.
For a stateless session bean, if you call the increment() method 5 times, it is not guaranteed that when you do a retrieve(), you'll get a 5. It is up to the container which EJB it'll assign to your call. So if you are assigned a new EJB instance, then you'll get a zero. It is also possible that the container has not cleaned up your EJB instance, so it might be possible to get a 5 -- but it is not guaranteed.
For a stateful session bean, if you call the increment method 5 times, when you retrieve the value you'll get a 5. The container guarantees that the EJB that was used the first time you called will be used for all subsequent calls.