Tangle is a "revolutionary new blockless distributed ledger which is scalable, lightweight and for the first time ever makes it possible to transfer value without any fees." It has been used on a cryptocurrency called IOTA.
Blockchain
On the Blockchain when someone sends a transaction, they sent it to a node ("node" has nothing to do with network nodes) that broadcasted it to miners, they verify if this transaction has taken place and bundle the result of this verification into a big database called the Blockchain.
Think of it as a student that does their math homework: they have to give their homework to their teacher to get a mark. The teacher check every answer to see if it is it correct or not and assign a grade to it.In this example, the miners are the teacher: they validate every transaction, and they get paid for it; that is their "job" on the blockchain, and that is why fees are needed: to pay those miners.
Tangle
Smarter teachers do one thing: if they do not want to do the work, they tell the students to mark each other's homework.
The IOTA ledger is constructed as a tree, and every transaction points to two randomly chosen previous transactions in the tree, and include a tiny Proof-of-Work hash. This creates an ever-growing tree representing the ledger.
Every time someone sends a transaction, each node verify other transactions in a sort of "network" of transaction verifications.
Because every node in Tangle do their calculation when they are sending their transactions, it results in a faster network without any need of fees because every node did the work already.
In other words: no one has to check no one's work because every node checks its own and no fees are needed anymore.
Transaction fees are mainly used to cover miners expenses. They also set a threshold on the minimum amount of payment below which money transfers become inexpedient.
No transaction fees means true micro and even nano-transactions.
Tangle does not seek to replace the blockchain, in fact, there could be Tangle-blockchain interoperability such as increases the security of blockchains by enabling the ability to include checkpoints for transactions.