Well, very, very, very, very dumbed down and simplified, the lightning network is basically something like this:

Person A opens a payment channel with person B. Person A and B funds the channel using an on-chain transaction of (for example) 10 BTC.
Person A can now send (for example) 20 offchain transactions of ~0.1BTC to person B (for example, person A has a ipTV subscription with person B, and he pays 0.1 BTC/month for his subscription, every month). Everytime, the status of the channel gets adapted with an offchain transaction.
If, in the end, the channel gets closed, the final status gets recorded into a transaction and broadcasted to the network.
So, in this case 18 on-chain transactions were avoided, only the funding and withdrawing transaction were on-chain.

Now, imagine person C. He opens a channel with person A.
He can now send 0.1 BTC to person B, since person A has an open channel with person B, and the channel is still funded. So in reality, person C sends 0.1 BTC offchain to person A, person A sends 0.1 to person B, so person B receives the 0.1 BTC that was sent by person C. Person's C available funds are reduced by 0.1 BTC, person A still has the same amount of funds as before the transaction, person B has 0.1 BTC more than before the transaction (fees were not taken into account with this very simplistic example)

When sufficient people have open (funded) channels with eachother, you can basically send funds over the lightning network to any person that has at least 1 or 2 open channels. Each node in the network can add a tiny fee, but it is also possible to minimise the fee by calculating the optimal path between 2 nodes/wallets/people.