Lightning Network (LN) is a network of payment channels which exists outside the blockchain. Lightning Network relies on you opening a two-way trusted payment channel with another user which then turns into a network.
The channel handles all the main transactions (opening and closing transactions) alongside all the secondary ones (all other transactions). All the main transactions are registered on the blockchain, but all the secondary ones exist only inside the Lightning Network payment channel that you have created.
Let's imagine this in real life. Suppose you come to the pub for a few beers. You know that you'll be staying in that pub for the whole night, so you open a tab (the channel), so that you don't have to pull out your credit card (slow blockchain wallet) to pay each time. It's quicker and easier that way. All the transactions on that tab (secondary transactions) happen outside your credit card (off the blockchain). But once you close the tab and pay with your card at the end of the night, the total you spent will be deducted from your card balance (main transaction registered on the blockchain).
Because of that, Bitcoin can now handle instant payments, can scale appropriately and have low fees.
Lighting Network was originally proposed by Thaddeus Dryja and Joseph Poon in a 2015 white paper, but active testing phase has only started at the end of 2017. Currently, there are already more than 1000 lightning nodes running on the network and that number is increasing.
Litecoin, Zcash, Ethereum and Ripple are also currently developing LN (or something very similar to LN) for their networks. Stellar has also expressed interest in the Lightning Network in order to improve security and increase the transaction speeds.