The TM1 REST API is not enabled by default, you need to update your tm1s.cfg on your TM1 server with the following parameter:

• HttpPortNumber=8881

The port number can be anything but not be used by any other application. You must make sure you use a different port number for each TM1 instance (server). 

You now need to restart the instance of TM1 for it to take effect using the Windows Services panel:

To see if the REST API is working open your favourite browser (my recommendation is Chrome) and enter the following URL (replace the port number with your one). It is important that you enter the URL exactly as it is below. Note the https as the protocol, TM1 uses SSL by default.

https://localhost:8881/api/v1/$metadata

Because the SSL certificate used by TM1 is for tm1server rather than localhost you will receive an error about your connection not being private (example of Chrome below). It is safe to ignore the error (in development only) so click on ADVANCED and then Process to localhost (unsafe). 

You will now see the content below, which is the TM1 REST API $metadata page that describes all of the objects / actions you can take against the REST API.