A way of structuring data, often used for data modelling, and increasingly as a consensus tool in cryptocurrencies.
Directed acyclic graphs consist of vertices and edges. There are no blocks, unlike in a blockchain. Instead, transactions are recorded as vertices, and these are recorded on top of one another. Transactions are submitted to the DAG by nodes, much like on a blockchain. In order to submit a transaction, a node must complete a Proof-of-Work task.
In a DAG, every new transaction must reference previous transactions in order to be accepted onto the network, in a similar manner to the way in which blocks on a blockchain contain references to previous blocks. When a transaction is referenced by another transaction, it is confirmed. In order for that transaction to be confirmed, it must be referenced by a further transaction, and so on.
DAGs have a number of perceived advantages. Transaction speeds are high as processing is not limited by block creation. There are no transaction fees because there are no miners — and, of course, the lack of mining has huge environmental benefits.
However, there are significant drawbacks too. Perhaps the most important is that they are not fully decentralized, unlike blockchains. In addition, the use of DAGs in cryptocurrency settings is very much in its infancy. These two factors combine to mean that, for the most part, DAGs are now used primarily as a way to get a network started, rather than as a system on which to build a stable and lasting network.