A .mts file is like a high definition mpeg2 file, which is the format used on standard DVDs.
A .vsp file is the project file created by your software it's not a video file format. You need to render the project into the correct format depending on what you will be viewing it with or on.
The standard Blue-ray disk format is .mts files.
avchd is a high definition video format is inside the .mts file.
There is a bit more to it than this but I don't want to confuse you by other information that you don't need to get your video onto a disk.