Warning: do not share your API key with anyone (including anyone at Airtable) since it's effectively a password that allows access to all your bases. If you accidentally reveal your API key, you should regenerate your API key as soon as possible at https://airtable.com/account.
To find your API key, navigate to your account page. On your account overview page, under the API heading, there's a button that says "Generate API key."
Clicking this button will generate an API key for you.
Once you've generated a key, you have the option of regenerating your key or deleting your key. To regenerate your key, click the Regenerate API key option. This will bring up a warning that regenerating your key will break your API integrations. Click the blue Yes, regenerate key button to confirm your key regeneration.
To delete your key, click the Delete key option. This will bring up a warning that deleting your key will break your API integrations. Click the red Yes, delete key button to confirm your key deletion.