Where does the mysqldump command store the backed up databases?
By default it does it to the working directory at the time the command was made. You need to redirect the output to a file if you need to save it.

The place you put the backup is up to you. Ideally, it should be on a different machine in a different physical location. This is why you usually do a remote MySQL dump over SSH and get the data on a different machine. You just pipe the console output over ssh into a file on a different machine.