Basically the Lightning Network is an upgrade to the current Bitcoin system (i.e., the protocol) that will allow users to have instant and near fee-less transactions, allowing efficient micropayments, a really required feature imho for long term survival of BTC.
It is a proposed solution for the Bitcoin scalability problem. The network will use this off-chain protocol and is currently under development, but Lightning Protocol Version 1.0 has been already released.
The specification for using the Lightning Network relies on Segregated Witness (SegWit), which was activated on mainnet in August, 2017 - as of block height 481,824.
Lightning Network could be used to send transactions across even different blockchains, like BTC <-> LTC, called atomic swaps.
I am quite a visual learner, let's watch some really useful videos:
Bitcoin's Lightning Network, Simply Explained!
Here’s a video by Andreas Antonoplos, which offers another very good explanation about the whole system:
Here is an video of Lightning Network being used:
When? In 2018 hopefully!