Here's a link to the Wikipedia artcle on the Lightning Network, henceforth refererred to as the LN. The LN is a mesh network on top of the Bitcoin or any other blockchain. It can be used to make peer-to-peer payments via cryptographically secure bidirectional payment channels. Opening a payment channel involves a funding transaction on the Bitcoin blockchain whereby a sum of Bitcoin is committed for as long as the channel is open. Off-chain transactions involving payments going back and forth can be done very cheaply and quickly until one or both parties decide to close the channel at which time only the final balance is committed to the blockchain. One advantage of the LN is increased scalability and the feasibility of micropayments. Micropayments that miners would never process otherways can be aggregated into their sums that are economically feasible to be processed by the Bitcoin blockchain. Another advantage is the possibility of cross-blockchain atomic swaps, that is, exchanging tokens of one cryptocurrency to another bypassing centralized exchanges.
The drawback of the LN is that it will not probably be a be-all and end-all solution to the scalability problem without sacrificing decentralization to a significant degree as the protocol forces the participants of a payment channel to commit currency worth the entire sum of the transactions for as long as the channel is open. This may necessitate the existence of central banking hubs for many use cases.
I think the introduction of the Lightning Network will help significantly. It would be very cool to use an AI-enhanced multiple-currency, combined fiat-crypto smart wallet on my mobile phone that knew my money traffic history in great detail and knew how to combine that information with my geographical location, travel plans and planned purchases of major items such as a house or a car. It would automatically open and close payment channels as necessary. If it knew I would be away from home for a week, for example, it would close the payment channels connecting it and the businesses that I frequent while at home (an operation that would take about 10 minutes on the Bitcoin blockchain) and open new ones based on expected behavior. Such a wallet would be a great tool for personal money management. It could automatically budget for fuel, loan payments, groceries etc. and provide detailed analytics at my request. Various businesses would be greatly incentivized to offer loyalty programs with benefits based on the amount of funds committed to their payment channnels integrated over time. OmiseGO is an Ethereum-based technology meant to be used hopefully in this manner. Ethereum will face the same scalability issues as Bitcoin should it be adopted by the masses. Ethereum has the Raiden Network, a scalability solution similar to the Lightning Network, in the works.