There is this cool trick you can do with Schnorr signatures. It is called Adaptor Signature (AS). An adaptor signature is an extra signature that, combined with the original signature, allow for revealing a value that was previously hidden. You can use this trick to solve trust problems as they appear in atomic swaps, coin swaps and Discreet Log Contracts (DLCs).

Signature Aggregation (SA) is a way to aggregate multiple signatures into a single signature. The single aggregate signature is smaller (in bytes) than the original signatures combined. It reduces transaction weight, meaning we can have more transactions per block, which is always a good thing. It's like 7zip for transactions. Signature Half Aggregation is a variant of SA that only aggregates half of each signature. It offers less compression, but it has the benefit of not requiring any interaction with the signers, whereas full aggregation does require cooperation of all the signers.

So two cool tricks, but the latter breaks the former when it is used for blockwide signature aggregation. This article explains the math behind it and why SA breaks AS

Lightning Network is a peer-to-peer payment network that runs on top of the Bitcoin Blockchain. Because it runs on top of the Blockchain it is called a layer-two solution, which groups it together with other solutions that have this property of being built on top of a Blockchain. Layer One is the base layer, where the Blockchain lives. Because of the inherent properties of Layer One, it is impossible to process large amounts of transactions in that layer. The Bitcoin Blockchain can famously process a mere seven transactions per second on average. This constraint was the main impetus for the creation of Layer Two solutions. Lightning Network, being a Layer Two solution, allows for near instant transactions, that can easily scale to millions of transactions per second.