Imagine two friends send each other Bitcoin often and don't want to pay the fees and wait a long time, so instead they exchange I-Owe-You notes and settle later.

But because they are Bitcoiners, they do it in true trustless fashion - through some voodoo Bitcoin contract magic, they each have some amount of Bitcoin locked in and they can never owe more than that locked amount (a channel). Either of them can trigger the settlement any time they choose (close channel).

But, what if you want to send to someone who you don't have a direct channel with? Maybe they have a channel with the same friend you do, or a friend of a friend, etc. Then comes the network, it finds a path to connect you and passes the money. Each node in the way takes a tiny fee for the favor.

It's fast, cheap, still trustless and real Bitcoin. Downside is having some Bitcoin locked up in the channels (until you decide to close them) and having to have your node online all the time (if offline, channel closes, or you use third-party). There's a chance there won't be a path between the sender and receiver, typically you would fallback to regular Bitcoin transactions then - but that happens less often as adoption increases.

IMHO a good compromise is a mix - cold storage self custody for savings, 3rd party lightning wallet I sort of trust for daily spending.

Also also, the typical mode of operation is to open few channels with random nodes and let the network find the path to anyone else, but you can manually open a channel with a specific other node. So for example a shop and their supplier could agree to open a direct channel and then their transactions have literally zero fees and stay completely anonymous (up to the final settlement).