Because the proxy is itself CORS-enabled, your application and the proxy don't have to be hosted on the same domain.
This proxy was tested with the Salesforce.com REST API, but it should work with other services as well.
There are different options to get your own instance of the CORS proxy up and running:
Deploy to Heroku (easiest): click the 'Deploy to Heroku' button at the top of this page
Install a local version
Clone this repository
Install the server dependencies
Start the server
Substitute the actual service URL with the Proxy URL
Set the request method, query parameters, and body as usual
Set the actual service URL in the url http://localhost:3000/YOUR_URL
Send the request as usual
The proxy allows all origins, methods, and headers. You probably want to lock this down in a production environment.
The proxy currently passes the "Authorization" header to the target endpoint. You can modify the proxy to pass additional headers (or all of them).
Check out James Ward's Saleforce CORS Proxy written in Scala.