Mailgun's API lives at https://api.mailgun.net/v3 (api.eu.mailgun.net for EU accounts). Sending is a form POST to your domain.
Mailgun uses Basic auth with the literal username api and your API key as the password — curl's -u, PostTaco's Auth tab. Find the key under API Security. In the examples it's written {{MAILGUN_API_KEY}}; {{MAILGUN_DOMAIN}} is your sending domain.
Full reference: https://documentation.mailgun.com/docs/mailgun/api-reference/
curl -X POST https://api.mailgun.net/v3/{{MAILGUN_DOMAIN}}/messages \
-u api:{{MAILGUN_API_KEY}} \
-H "Content-Type: application/x-www-form-urlencoded" \
-d "from=PostTaco <postmaster@{{MAILGUN_DOMAIN}}>" \
-d "to={{TO_EMAIL}}" \
-d "subject=Hello from PostTaco" \
-d "text=It works."
Deliveries, opens, bounces, and failures.
curl "https://api.mailgun.net/v3/{{MAILGUN_DOMAIN}}/events?limit=10" \
-u api:{{MAILGUN_API_KEY}}
curl https://api.mailgun.net/v4/domains \
-u api:{{MAILGUN_API_KEY}}