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From Ethereum White Paper:
"What Ethereum intends to provide is a blockchain with a built-in fully fledged Turing-complete programming language that can be used to create "contracts" that can be used to encode arbitrary state transition functions, allowing users to create any of the systems described above, as well as many others that we have not yet imagined, simply by writing up the logic in a few lines of code."
How can you explain it in non-technical words?
Let us start with the concept of a file with contents, for reference, because I think we can all relate to what is going on in that case.
In this context, the contents are the condition of the file, a.k.a. the "state" of the file. The file's state might change from time to time as someone or something edits the file. Each revision is a transition because someone did something.
Now, let us introduce some blockchain concepts There is no file, there is a blockchain that contains some data. It is always evolving because there are always new transactions that cause the state to transition from an old state to a new state. If Alice had 10 Eth and Bob had none, then Alice sent 3 ETH to Bob, then the state transitions from the old state to a new state. In the new state, Alice has 7 and Bob has 3.
Such simple transfers are protocol-level state transitions with protocol-level rules. Such rules are not arbitrary. The rules say Alice cannot send more money than she has, she has to sign her transaction a certain way, and so on.
"Arbitrary" state transitions is about contracts that work with data (contract state) that has meaning in the context of the contract. In this context, arbitrary means the developer is free to define the layout and interpretation of data within the contract's own storage space. The developer can define the functions and the rules enforced by the functions, strictly constraining state transitions to any set of rules the developer deems appropriate. The rules governing changes to the state of the contract are arbitrary.
Hope it helps.
