1)
If you trust cloud storage, it does seem like the best option. If done right, it's protected against failures of the vendor's systems by having multiple copies of the data, and you can get it anywhere. But the network is a slow way to do the initial full backup, so be sure that the setup does incremental updates after that.
2)
If you are paranoid like me and don't want to give all your information to someone else on general principle, cloud is not so good. Microsoft's service doesn't even encrypt your data at rest. So my favorite backup drive is: Any 3.5" hdd. I've got two hot-pluggable bays in my machine. I just grab an HDD out of my antistatic, shock-resistant drive case (I told you that I'm paranoid), slip it into the bay, and do my backups. I do restorable image backups of my system drive and file backups of the others.
3)
Always have more than one backup drive. When you do a full backup, choose the one with the oldest backup. If the system fails catastrophically while doing the backup, you were not in the process of overwriting your last backup!
4)
Ideally, you should make two copies and keep one somewhere else. A bank vault, the home of a family member, Iron Mountain.... But if, God forbid, something happened to your home so that your computer and your backup drives were totaled, the backups might not be the most important thing on your mind.
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External drives with USB 3.0 or Thunderbolt or LightningBlast are great devices, but I would only use them for a mobile PC. When I build a home system, I leave room for a hot-plug bay. So much cheaper to have multiple backup drives that way. If there's no room, build in an eSATA port and use an external docking bay.
@funkytwig - if you are running continuous backup, I would agree with you 100%. It's something I'd like to try some day.