OBS Studio (a rewrite of the original Open Broadcaster) is a comprehensive open-source package for video recording and live streaming.
The program supports multiple sources, including media files, games, web pages, application windows, webcams, your desktop, microphone and more.
These sources can be combined. For example, you could include a web development tool, a browser window with a live preview of the site you're updating, and a microphone to tell the viewers what you're doing.
A Record option can save your show/ demonstration/ game/ presentation as an FLV/ MP4/ MOV/ TS/ M3U8 video with your choice of encoder and settings.
Alternatively, there's support for live streaming via Twitch, YouTube, hitbox.tv, DailyMotion, beam.tv, Livecoding.tv, Facebook Live or Restream.io.
The program doesn't make much of an effort to lead you through all this, but it doesn't take long to figure out the basics. Add a source, explore the settings, click Start Recording and check the video results.
This is the portable version of OBS Studio.
OBS Studio could be easier to use, but you'll soon learn the basics, and overall it offers plenty of video recording/ streaming power.