What is the difference between centralised, decentralised, distributed, fully-distributed, partially-centralised, partially-decentralised system? Which type or topology is the system attached in the picture below? For instance, A is a master node (coordinator) of A1, A2, A3, which makes this part a centralised system. A, B, C and D are all connected between each other, which makes this system a decentralised one. But then, what are fully-distributed, partially-centralised, partially-decentralised systems?
Centralization tells you where the processing gets done.
Fully centralized systems do all their processing in one place.
Distribution has nothing to do with processing. It just relates to systems that communicate via messages.
A non-distributed system is a stand-alone PC with no clients, or a server with dumb clients.
A fully distributed system does not have a single source of truth, however that does not mean that all nodes have to be equal. You can have 'supernodes' mixed with lesser nodes and still be fully distributed.
Distribution = network
The current incarnation of distributed systems are networked systems, but of course it's perfectly possible to pass messages using a serial cable, sneakernet or carrier pigeon. Is that a distributed system?
Yes if both systems do actual work, No if one system is simply a recipient of data (printer/terminal etc).
Peripherals do not count
The passing of screen and keyboard data between the terminal and the server is not deemed substantial enough to qualify as message passing.
The definition of distributed computing is best narrowed down in the context of a specific protocol or algorithm (e.g. bittorrent or mapreduce)
Outside that context it becomes exceedingly vague to the point of being meaningless.
The mysterious diagram
The diagram in the question seems to hint at some sort of hierarchy and thus centralisation.
Obviously the systems A,B,C and D cannot be peripherals (because they are not leaf nodes) and thus some message passing (aka network activity) ought to take place.
What the nature of the leaf nodes (A123 et al) is is unclear, but these leafs do seem subservient to A,B and C proper.
In addition to Johan's answer:
A decentralised system is a fully distributed system where all the nodes in the network are equal so that there is no "authority" node in the network. Bitcoin Network is an example of this.