Postmark's API lives at https://api.postmarkapp.com. Each server in your account has its own token, sent in the X-Postmark-Server-Token header.
Copy a server's API token from its API Tokens tab in Postmark. In the examples it's written {{POSTMARK_SERVER_TOKEN}}. The literal value POSTMARK_API_TEST works for trying the send call — it validates without delivering.
Full reference: https://postmarkapp.com/developer
From must be a verified sender signature.
curl -X POST https://api.postmarkapp.com/email \
-H "Accept: application/json" \
-H "Content-Type: application/json" \
-H "X-Postmark-Server-Token: {{POSTMARK_SERVER_TOKEN}}" \
-d '{"From": "{{FROM_EMAIL}}", "To": "{{TO_EMAIL}}", "Subject": "Hello from PostTaco", "TextBody": "It works."}'
curl "https://api.postmarkapp.com/bounces?count=10&offset=0" \
-H "Accept: application/json" \
-H "X-Postmark-Server-Token: {{POSTMARK_SERVER_TOKEN}}"
curl https://api.postmarkapp.com/stats/outbound \
-H "Accept: application/json" \
-H "X-Postmark-Server-Token: {{POSTMARK_SERVER_TOKEN}}"