Here, in brief, are your options:
Backup to a reliable online service
Pros: Automatic, completely offsite, server is backed-up frequently, files can be easily accessed from a remote location after a disaster, and it’s free or inexpensive if you have a limited amount of data.
Cons: Uploads of large files can be slow (run backups at night), and relatively costly if you have a lot of large files like videos.
Backup to portable hard drive
Pros: Fast, easy, large amount of storage space, relatively inexpensive.
Cons: Must be taken offsite; subject to physical damage; vulnerable to physical, electrical or magnetic damage.
Backup to thumb drive or memory card
Pros: Fast, light, easy to take offsite, less vulnerable to physical damage (dropping) than hard drive.
Cons: Storage constraints for very large files, more costly per GB than hard drives, and small size makes these devices easier to lose.
Backup to optical media (CD or DVD)
Pros: Cheap, easy to mail to another location, not subject to damage from magnetic sources, relatively durable, easy to store.
Cons: Relatively slow, limited storage space per disk.
Backup to another computer on a network or to a network hard drive
Pros: Fast, easy.
Cons: Other computer or drive is likely to be in the same location and vulnerable to the same disasters.