Before to get a video list from a youtube channel or an playlist Youtube had a simple resource through your API V2, but now the version 2 was dropped and we have new V3 now, if you use the V2 yet you must to have an error like this:
{"apiVersion":"2.1","error":{"code":410,"message":"No longer available","errors":[{"domain":"GData","code":"NoLongerAvailableException","internalReason":"No longer available"}]}}And the new Version 3 API need to use oauth to to authenticate and all is a little more complex, but we have yet a simple method to get a video list from an user or one playlist using XML RSS format, for example to get a video list from my youtube user:
https://www.youtube.com/feeds/videos.xml?user=Richzendy
And to get an playlist:
https://www.youtube.com/feeds/videos.xml?playlist_id=PLYvJ3ZpD6FKIDaoe_uVXHk_2W1Bqg54tv
But to this can be usefull on a webpage you need to process the XML data and extract the important fields, to make this i created a little php script (you can adapt to you need), for now i get from youtube: title, video url, thumbnail url and description.