Lightning Network Invoices
Receiving payments through invoices is one of the most common activities for Lightning applications. You're most likely already familiar with receiving payments via Bitcoin through an address. The Lightning Network handles payments in a different manner. The primary mechanism for receiving payments is through an invoice, also known as a payment request. In the most common use case a payment request is generated by the recipient and is provided out-of-band to the payment sender. The request is a one-time use thing that expires after some duration.
For example, Alice runs a web store and Bob wants to buy a t-shirt. He adds the shirt to his cart and goes to check out. At this point, Alice creates an invoice for Bob's purchase. This invoice includes the cost of the shirt, a timeout that Bob needs to complete the transaction within, the hash of a secret value generated by Alice, and Alice's signature denoting that she indeed created the payment request.
Based on the information encoded in the invoice, invoices are typically one-time use and are intended for a specific purpose and amount. Functionally, this means that an invoices tells the sender: who, how much, and within what time frame to send a payment. The invoice is also digitally signed by the recipient. The signature ensures that an invoice can't be forged (Carol can't create an invoice for Alice). The last and possibly most important piece is that the invoice includes the hash of secret information. This hash obscures the secret information that will only get revealed once a payment is made.
So when Bob pays the invoice and Alice receives the payment, she reveals the secret. Revealing this secret acts as a proof of payment. Alice would only ever reveal the secret if Bob has made payment. Bob can only possess the secret if Alice gives it to him. Bob also has a signed invoice from Alice stating the conditions of the transaction. So once Bob pays Alice and she reveals the secret, Bob has a signed message from Alice and the secret that he can use as proof of payment.
So why all this complexity?
It enables one of the primary purposes of the of the Lightning Network which is trustless payment flow. This scheme allows payments to flow through the network even if Bob and Alice aren't directly connected. If you're unsure on how this works or want a refresher, I recommend reading this article on HTLCs payments.
For a more thorough walk through of invoices, check out Chapter 15 of Mastering the Lightning Network by Antonopoulos et al.