In this case the observer is only able to view their own cards and potentially those of the dealer.
A memory system can be used to remember the previously dealt cards that are now on the used pile.
This adds to the total sum of knowledge that the observer can use to make decisions.
In chess (apart from the 'who is moving next' state, and minor subtleties such as whether a side has castled, which may not be clear) the full state of the system is observable at any point in time.
Partially observable is a term used in a variety of mathematical settings, including that of artificial intelligence and partially observable Markov decision processes.