This is very similar to: Get URL/Embed code to private Vimeo videos programatically
Unfortunately the answer there has not been marked as the answer, a necessary step to link the two questions together, so I'll repost it here.
- Register an API app on https://developer.vimeo.com/apps
This is necessary for every API app. We need to know who is using our system, and how to contact them if necessary.
- Generate an access token.
There is general documentation at https://developer.vimeo.com/api/authentication, but you will likely be using the "single user application" workflow. It's a lofty title for "generate an access token via the UI on your application page, then hard code it into your app". This access token will interact with the API on behalf of the user who registered the application.
- Request your video information.
There are many different API calls to get video information. You can find these at https://developer.vimeo.com/api/endpoints. /me/videos will show all of the authenticated users videos, /videos/{video_id} will show a single video.
One extra note, if you are using PHP you should use the official Vimeo PHP library: https://github.com/vimeo/vimeo.php