The Lightning Network is a proposed capacity upgrade for Bitcoin, which would allow it to scale on a planetary level. It works by creating a peer to peer mesh of payment channels. The payment channel technology was proposed by Satoshi Nakamoto and other reputable people, but the most advanced implementation is called Hashed Timelock Contracts (HTLC).
Over time, there have been several proposals, such as Blockchain.info's Thunder Network. Due to the transaction malleability bug, this implementation would have risked the participants money, and an evil actor could have stolen funds from channels. Only trusted peers could have established payment channels. Thankfully, due to Bitcoin's 2017 Segregated Witness upgrade, this is no longer a possibility, and cryptographically secure payment channels can make the Lightning Network real.
In the lightning network, payments are onion-routed (yes, just like TOR) via already established channels, so in order to use it, you need to create a connection to a peer with an on-chain transaction. Afterwards, you can make as many payments as you want, as long as you don't exceed the channel capacity. Any old bitcoin transaction could be replaced with an channel open + pay, if both parties agree to this, so the Lightning Network could grow on-the-go. When you want to stop using the Lightning Network, you can close the channel, and the necessary funds will go into a bitcoin address that you control.
You can test the Lightning Network on the Bitcoin Testnet using HTLC.me's custodial wallet, or if you want to get your hands dirty, you can set up c-lightning Eclair or lnd. The lightning-app seems to be the only desktop app which you can use without running a full bitcoin node (which requires some bandwith and free HDD space), because it uses a light client called neutrino.
If you can't route the payment, open a channel to us, we are firstname.lastname@example.org on the Bitcoin Testnet.