Thank you for your interest.
I try to retrieve tweets from a search (no problem for this part in JSON call and response, this seems to me clear).
I tried with examples provided by CodeBird JS library (I have put my public and secret keys to my application).
Something wrong authentication with OAuth. In the Application Management I defined website and callback fields (and without). A popup asking me to authenticate and to accept that my application access to my data (read-only): I accept then told me to return to https://apps.twitter.com to put a code (seems a number or a token I do not know).
I see in my permission tab that I can revoke my app then I presume it has rights, but each time I launch my page authentication popup give me a code to past in this https://apps.twitter.com (but where exactly).