I'm creating an Aura app and I can't find anything that says what database is being used. I know Salesforce provides a database using a Salesforce org. What database software is used when hosting a stand alone Aura framework app?
You need to first understand the basic of Salesforce and they way it works. Here is key differences of different keywords you have added in your question
Aura is a UI framework for developing dynamic web apps for mobile and desktop devices, while providing a scalable long-lived lifecycle to support building apps engineered for growth. It supports partitioned multi-tier component development that bridges the client and server. Salesforce use that framework to build Lightning component and apps.
Salesforce database: Force.com combines several different persistence technologies, including a custom-designed relational database schema, which are innately designed for clouds and multitenancy. It means there is no such term called database for every single tenant. They share common database with each other. Read this article for more detail about Salesforce multi tenant architecture
Hosting: all application hosted in force.com platform in Salesforce so if you want to use your app for your personal use you can signup for a developer edition account and create your app. but if you want others to download and use your created app. You can develop your app. Signup for partner account with Salesforce and host your app on app exchange which is marketplace of Salesforce enterprise app.