Link to oficial API documentation is http://instagram.com/developer/authentication/
Longstory short – two steps:
Get CODE
Open https://api.instagram.com/oauth/authorize/?client_id=CLIENT-ID&redirect_uri=REDIRECT-URI&response_type=code with information from http://instagram.com/developer/clients/manage/
Get access token
curl \-F 'client_id=CLIENT-ID' \
-F 'client_secret=CLIENT-SECRET' \
-F 'grant_type=authorization_code' \
-F 'redirect_uri=YOUR-REDIRECT-URI' \
-F 'code=CODE' \
https://api.instagram.com/oauth/access_token