Skype is often the goto application for videoconferencing (Google Hangouts is also great) but there are alternatives.
Zoom is a free, easy to use cross-platform application that prompts recipients to download a small app, although you can just use their web client, albeit with reduced functionality. You can also record up to 40 minutes and download the MP4, host up to 50 participants, and hold an unlimited number of meetings.
You can also videoconference people via the popular Facebook messenger platform through the web interface or mobile apps, and you don’t need to be ‘friends’ on Facebook to do so. You will be prompted to add them as a contact, but this will not enable them to see any more of your personal profile.
If you want to use Skype or Google Hangouts, you can use a free service like Free Video Call Recorder to capture your videoconference.