You need the Manage System Settings permission to do this.
Go to Settings → Business Settings → Integrations → API Keys.
Click + New Key.
Type a Label so you can tell the key apart from others, such as the name of the app that will use it. Then click Create.
Copy the key and keep it somewhere safe. It's only shown once. If you lose it, you'll need to create a new key.
Good to know
You're charged for each API request made with your key.
To stop a key from working, click Revoke next to it in the API Keys list.