IN PLAIN ENGLISH:
Lightning Network allows two parties to conduct multiple transactions between themselves without having to record every transaction immediately to the blockchain. Instead, the Lightning Network works by creating a direct channel between the two parties where they can do multiple transactions over a decided amount of time, and then only post the transactions to the blockchain once all transactions are complete. This means paying one transaction fee rather than multiple, and reducing the load on the blockchain network.
The details:
Lightning Network works by creating a Layer-2 channel between two parties and using multi-signature Smart Contracts and time-locks to guarantee payment. This channel works on top of the blockchain so there are no fees and transactions between parties are instantaneous. A transaction fee is only paid once the channel is closed and the transactions are recorded on the blockchain.
For example, say you have a babysitter that wants to be paid in Bitcoin. As it is currently, every time you pay her, the Bitcoin you send will be recorded as a new transaction, it will take time to be confirmed and cost a fairly substantial transaction fee. What Lightning Network allows you to do is create a channel off the blockchain, directly between you and the babysitter.
Let's say that you agree to pay her $200 per week for her services. You can create a payment channel with her that says the channel will be open for a period of 4 weeks, and you will pay her $200 per week for those 4 weeks. Every week you send $200 to the multi-sig address with your signature on it, and it sits there waiting until the babysitter adds her signature to receive it. At the end of the 4 weeks, you have sent 4 transactions of $200 each and the babysitter can add her signature to the transactions to receive the payments. At this point, once both signatures are added, the payment channel is closed and the transaction is recorded to the blockchain. So rather than paying 4 transaction fees to pay the babysitter, only one transaction fee was charged at the end.
As more people use Lightning Network, payment channels can be made between two parties that don't know each other through a mutual third party that they are already connected to, rather than creating a new channel between each other directly.
This technology will allow mass adoption of Bitcoin and other cryptocurrencies, and allow Bitcoin to solve a lot of the issues it is currently facing with high fees, slow confirmation times and trouble scaling. Bitcoin is planning to implement Lightning Network in the near future, and Litecoin has already added it.