How to connect to REST API from remote host?
To get the rest API working, you need to call an url looking like that (this example to create a contact)

http://www.mysite.com/wp-content/plugins/civicrm/civicrm/extern/rest.php?entity=Contact&amp;action=create&amp;api_key=(myAPIkey)&amp;key=(mySiteKey)&amp;json={“contact_type":"Individual","first_name":"Tester","last_name":"Yolanda"}


The specific path to rest.php depends on what CMS you use

A few pointers:


you need to set the key with the site key (from civicrm.settings.php) 
you need to create an api key on the contact that peforms the operation, and
this contact needs to a user that has the appropriate permission (start with an admin account to rule out the permission issue)


This being said, your error message is not normal, you should get a "Failed to authenticate key" error message

https://civicrm.org/sites/all/modules/civicrm/extern/rest.php


it might be that you don't use the proper url to reach the rest.php file, or that something needs to be configured on your webserver so it's properly processed

What is the url you try? it's a different one from the ajax api used by the api explorer

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