You can apply for a personal or production app by clicking "Register Project" on the main dev portal page.
Do not apply for a project if you just want to test the API. If you are developing a project or just testing things out, use your development key that Riot gives you automatically when you signed up. This key expires every 24 hours, so you need to refresh it every day that you work on your app. This is deliberate functionality and your project will not be given a non-development key if all you are doing is testing the API. Even the Rioters use development keys that expire every 24 hours.
Applications take ~ two weeks (10 business days, barring holidays) to process. Make sure it's verified (riot.txt). If it's been longer than two weeks and you didn't change your application during that time period, ask a Guru on Discord for help.
You can find your app ID by going to https://developer.riotgames.com/apps, selecting your project from the list on the left, and copying the number in the URL (https://developer.riotgames.com/app/YOURAPPIDISHERE/info).
Register your project to apply for a Production API Key with access to the Standard APIs and/or Tournaments API. A Production API key requires a working prototype. Production API keys are not for testing purposes. You may submit an application in the planning stage of your project, but your application for a production key won't be approved until your project is ready for public consumption. Work in progress should be tested using your demo key.
Register your project to apply for a Personal API Key without the verification process. This is only for the Standard API's, not Tournaments. Personal Keys require a detailed description ofthe project.