I want to record a video and want to play some recorded voice over the video, but I still want the sounds of the original video to remain. Is this possible, and when the answer is yes, can you recommend the software you are doing this with?
It is possible in almost any video=editing software.
If you are using Windows and you do not have much experience in video editing, than you can make it Windows Movie Maker. It is free and simple for beginners.