Jenkins has a link to their REST API in the bottom right of each page. This link appears on every page of Jenkins and points you to an API output for the exact page you are browsing. That should provide some understanding into how to build the API URls.
You can additionally use some wrapper, like I do, in Python, using http://jenkinsapi.readthedocs.io/en/latest/
Here is their website: https://wiki.jenkins-ci.org/display/JENKINS/Remote+access+API
~ Answered on 2014-09-05 12:28:46