This is very easy with most video player software included with your computer.
In Windows 10, your computer comes with "Movies and TV" app, which can easily do this for you. Follow these steps:
- Copy all your video files to a single folder
- Rename them so they alphabetically sort in the order you want them to play
- Highlight all, right click on one of them, select open. From there, the first video will start playing. If a different player is your default player, it will open in that instead. Just change your default player before this step.
- To set a video on repeat, click the menu button in the bottom right corner, then click Repeat.
- You can advance to the next video when you are ready by clicking the advance arrow that overlays the right side of the video.
- When you advance to the next video, you have to set the repeat again or it will auto-advance to the next video.
Other players like WMP, VLC, and iTunes will all do this about the same, but may have different ways that you would have to set it up. For example, WMP and iTunes will only play multiple files in succession if you add them to your Library first, then a playlist.
All three of those are free, so you can fiddle with them and see what you like best.
For a robust professional experience, you need a software that offers "instant replay". My first thought would be vMix, but it's pretty pricey and more geared toward capture, though it has a "replay" function. These kinds of software will show you all the videos you've imported for the instant replay and have all kinds of functions like repeat and advance. They are designed to do exactly what you are looking for, but none of them are free.