Instant Payments. Lightning-fast blockchain payments without worrying about block confirmation times. Security is enforced by blockchain smart-contracts without creating a on-blockchain transaction for individual payments. Payment speed measured in milliseconds to seconds.
Scalability. Capable of millions to billions of transactions per second across the network. Capacity blows away legacy payment rails by many orders of magnitude. Attaching payment per action/click is now possible without custodians.
Low Cost. By transacting and settling off-blockchain, the Lightning Network allows for exceptionally low fees, which allows for emerging use cases such as instant micropayments.
Cross Blockchains. Cross-chain ATOMIC SWAPs can occur off-chain instantly with heterogeneous blockchain consensus rules. So long as the chains can support the same cryptographic hash function, it is possible to make transactions across blockchains without trust in 3rd party custodians.
See Reddit megathread for large number of linked articles. 
SecurityEditVideo from Andreas Antonopoulos on the Lightning Network and onion routing within it.
- https://arstechnica.com/tech-policy/2018/02/bitcoins-lightning-network-a-deep-dive/ Ars Technica 2018-02-04 explanatory article
- https://www.youtube.com/watch?v=c4TjfaLgzj4 2018-02-15 Andreas Antonopoulos Bitcoin Q&A: Misconceptions about Lightning Network
- https://medium.com/@menirosenfeld/a-flash-of-insights-on-lightning-network-338aea52e2bcA 2018-03-13 Flash of Insights on Lightning Network