An app that would be installed on a merchant's shop to provide some benefit for them (usually in exchange for money). They can be custom apps for a single shop or a public app on the Shopify App Store.

Yeah I’m confused too, I thought Shopify was a platform like Wix where you could only customize the UI. Are these apps for hosted installations where you can run arbitrary code?

They are apps that use Shopify APIs to add features. They don't run on Shopify servers, but talk to them through the API, can embed UI in Shopify backend dashboards, ...