Cryptoeconomics - sounds like something to do with how cryptocurrencies are spent and the economies that exist around cryptocurrency, but it isn't quite that.
Cryptoeconomics is the combination of cryptography and economic theory to create sustainable operating protocols for trust-less and decentralized platforms. Cryptoeconomics is how to get all the participants on the blockchain - the miners and the entities transacting - to act in ways to reduce the likelihood of unfavorable behaviors that could happen on the blockchain that would impact other actors.
Cryptography allows for the passing of messages in a secure way. By obscuring a third party from seeing messages passed between two parties (Alice and Bob are the names of the two parties passing messages in the tradition of literature written on cryptography, with the eavesdropper being named Eve), peer to peer interactions of all types become secure. Layering economic incentives on top of the secure protocols compels users of a platform to act in certain ways on the platform because of rewards or penalties tied to their behavior.
Solving the Double Spend Problem
Let's look at the Bitcoin blockchain. To remind you, Bitcoin is a network for peer to peer financial transactions.Two individuals, Alice and Bob, want to transact with each other. Alice decides to send some money to Bob in order to pay for a bicycle he sold to her. In order to send Bob the money, Alice signs her transaction with her private key. Bob can use her public key to verify that the transaction is from Alice. Thus, cryptography is used in the transaction in order to prove that Alice owned the coin she is transferring to Bob. In order to prevent Alice from spending that same coin again, an economic incentive is created within the network for others to check for a double spend, without needing a central authority (a bank or government, for example) to do this checking.
On the Bitcoin blockchain, individuals running a node will compete to combine the transactions that are sent out to all nodes into a set block, and if they are successful in solving a hashing problem (once again, using cryptography) that is accepted by the other individuals running nodes, then they are given a reward - in this case a set amount of Bitcoin for their troubles of using their computing power to create the next block in the blockchain (the specific details of the Bitcoin blockchain when it was created can be found in the Satoshi Nakamoto’s “Bitcoin: A Peer-to-Peer Electronic Cash System”). If they were allowed to double spend, and this is caught by others who are verifying the transaction, and it would not be accepted as a block in the chain. Thus, every miner is incentivized to ensure that there is not a double spend of Bitcoin.
You can see how cryptography, as used in peer-to-peer transactions and the creations of the blocks, is combined with an economic incentive to compel actors on the network to behave in a way that ensures spending of tokens, work required to sync the ledger, and verification of transactions is done in such a way that it meets the expectations of those using the blockchain. Specifically, as illustrated in our example above, some of the problems that crypto economics allows us to solve in blockchains deal with malicious actors. Others deal with how we can create value on the network, or incentivize entities or individuals to use their computing power in order to sync the ledger, store data, provide other entities with data, or do a host of other activities.