The Lightning Network is a system that is being built on the basis of Bitcoin blockchain which enables quick and cheap payments directly between two parties. To achieve this quick and cheap transaction, the following steps are taken:

A multi-signature wallet that holds some amount of bitcoin (provided by at least one of the two sides) is set
The wallet address is then saved to block Bitcoin public including a balance sheet (smart contract) which proves how much this bitcoin deposit is owned by
Once this payment channel is set up once, it is possible that both parties make unlimited transactions without ever touching the information stored in the blockchain.
With each transaction, both parties sign an updated balance sheet to always reflect how many bitcoins are stored in the multi-sig wallet of whom
The updated balance is not uploaded to the blockchain but both parties keep a copy
Whenever a dispute or payment channel closes, both parties may use the last signed balance sheet to pay their share of the multi-sig wallet
This all sounds very inconvenient, but actually for end users there is almost no additional effort required to make a Lightning payment, all of the above will happen automatically in the background.