I'm working on a professional project that I need to apply a watermark to a video.
How is this done? Are there apps that will do this free or low cost?
You do it in your video editing software. It's basically just a graphic like any other text you'd put on it. You just add it as a layer and position it where you want and for how much of the video you want it to show.
For example, my copyright watermark I put on my Iceberg flood video looked like this in my editor:
That purple bar on the top timeline is the copyright text that I watermarked the video with. Mine was text but you can do the same with an actual graphic like a JPEG or PNG. A PNG is usually better if it's setup for transparency. You can also adjust opacity of your graphic so that it appears to be slightly see-through like a watermark. With mine, I left it white with a drop shadow so it stands out on light or dark backgrounds.