A camcorder combines a video camera with a video recorder. Cell phones and digital cameras (point-and-shoot or D-SLR) can also record videos, however in a camcorder recording videos is its primary function.
Video cameras have made a huge progress in the past 20 years. Today most of them use flash memory and the quality of of the footage on high-end models is getting close to those you can see on HDTV.
One of the key advantage of dedicated video cameras over smartphones and point and shoots is that they usually provide a good optical zoom, which allows you to record distant action. The optical zoom range is probably one of the first things you should watch out for. Remember not to confuse optical zoom with digital zoom, as the latter simply zooms in the image in the center rather than giving you any extra detail.
Electronic image stabilisation comes as a standard in most models, however optical image stabilization is what you need if you want to shoot steady movies at full zoom.
Full HD capability doesn't automatically guarantee high quality of your footage. The lens and sensor are far more important here. However, with Full HD 1080p monitors and tv sets widely used today and 4K standard getting more and more popular, you shouldn't really be investing in anything that has video quality of less than 1080p.
Many modern camcorders today ditch the rear viewfinder in favor of a large LCD screen. Its large size as well as high resolution (at least 260,000 dots) will help you frame your videos more easily.
Even if you don't plan to take advantage of more advanced features from the beginning (e.g. adjustable exposure), it might be a good idea in having them in case you need them in the future.
For example, you might realize that you built-in microphone doesn't meet your expectations. In this case you will need an external mic input, and that's something you should look for in your product research.
Below we include a list of a few models you might find interesting: