There's a vast amount of documentation available, but all of it assumes a lot of prior knowledge or very particular use cases. Here's a simple and straightforward guide to creating and using a service principal for an Azure Key Vault so that your secrets can be managed programmatically.
Please note: I'm excited because I've finally managed to authenticate using a secret, it's probably more secure to use certificates but I'll get to that another time.
Step 1: Registered App
Under Azure Active Directory in the Azure Portal, select App registrations.
Add a New application registration, the application type being Web app / API and the Sign-on URL anything being any valid URL (just the format, it doesn't have to exist). If the name you enter isn't simple to remember then it would be a good idea to take a note of it for step 2.
Take note of the Application ID as that will be your Client ID for authentication, then select the Keys blade under Settings. Enter a Key description (preferably indicating the user or application that will be using this key), select a duration and Save. Immediately store the resulting value somewhere safe as it will never be displayed again.
I recommend adding yourself as an owner on the Owners blade (also under Settings), whatever else this may be good for it'll let you see the app registration immediately on the App registrations blade without having to select "All apps".
Step 2: Key Vault permissions
Open the Key Vault in the Azure Portal and select the Access policies blade under Settings. Click Add New and click on Select principal - you'll have to enter the full name of the registered app you created in the previous step in the search box before it'll show up, at which point you'll be able to select it.
You can either select an appropriate template from the top dropdown or choose Key, Secret or Certificate permissions manually. Don't worry about Authorized application at this stage.
IMPORTANT: pressing the OK button will add your new policy to the list, but it will not be saved! Be sure to click Save before continuing.
Step 3: Accessing the Key Vault from your Code
There are many different ways to authenticate, most of them obscure and undocumented. This is the simplest method, I've put the credentials in the code for clarity but I have faith that you'll store them somewhere more intelligent. Never store credentials in the codebase. Seriously. Just don't.