Software? A simple script, started using crontab, would be my approach.
# save current date (read man date if you want to add time as well)
#create backup using mysqldump
mysqldump --opt yourdatabasename > /path_to_directory/yourdatabasename.$mytime.sql
# mount your removable disk (read man mount)
# copy to removable disk
cp /path_to_directory/yourdatabasename.$mytime.sql /path_to_mountpoint
# unmount removable disk (read man umount)
If the database is big, you might want to (tar and) zip it during the copy (replace the line containing cp by the line below)
tar -cvf /path_to_mountpoint/dbbackup.$mytime.tar.gz /path_to_directory/yourdatabasename.$mytime.sql
I have not tested it. It should create a database backup (not zipped) including a date in the filename and copy or tar it to a mounted removable disk.
If the database requires a username and password for access, you have to do some reading (man mysqldump; options -u and -p)