To answer your main question: you should use both, a local and an off-site backup.

The problem with media files: they usually are big and a pain to back up to an online service (it may take several weeks to upload and days to download). On the other hand, there's no need to back them up hourly or daily. You will save your pictures and movies just once. Please make sure you save and back them up in a format that is not dependent of a particular software and choose a format and backup media that most likely still will be around five years from now.

Off-site backup is crucial. I already know several people who lost their data AND local backup at the same time through theft/burglary, lightning, or sabotage.

Off-site backup in the cloud may be very useful for less obvious reasons, too: your data is instantly accessible from anywhere in the world for whatever reason. You may be abroad using a freshly purchased computer. Now you can feed it with some of your data.

Off-site Backup solutions:

very easy:
Mozy (for OS X) http://mozy.com
Carbonite (for Windows, soon OS X) http://carbonite.com

also convenient, very cheap and safe (allows own encryption):
JungleDisk in connection with Amazon S3. http://jungledisk.co

You mentioned Drobo for local backup. I LOVE my Drobo. It really works as advertised. I'm using the 2nd generation Drobo as an NAS (Network Attached Storage) with an Ethernet DroboShare. Of course, I would prefer a Drobo with a real Ethernet connection (instead of the USB get-around) but the DroboShare does its job, too. By the way: if you already own an Apple Airport Extreme base station, you don't even need the DroboShare.

However, a Drobo is not exactly an ideal backup drive. I'm using it as primary storage and as a backup device, but I make sure that I have an additional backup of important data somewhere else. A Drobo is a great storage solution and fast enough to stream videos through your LAN. I have my whole music and media library on my Drobo. And I put a copy of everything important on my Drobo, too. But because a Drobo is as convenient as a simple external hard drive it's also very easy to accidentally delete files or backups. That's why I really would recommend having an additional backup solution, too.

A very convenient, cheap and fast solution for local backup are those "Hard Disk Toasters" or "HDD Docking stations" which allow inserting a 'naked' hard disk drive like a toaster. They connect via USB or E-SATA and using E-SATA, they will be as fast as an internal hard drive. This allows you to buy inexpensive 'naked' hard disks and use them as a fast, safe and convenient data storage medium.

Both.

I use a drobo. And I use mozy.com. You should keep an extra local copy for simple hardware failures. And you should keep a copy elsewhere in case of catastrophic things like a fire at your house.

I would say that depends on how often you plan on backing up, and how critical the data is. Consider at least the following scenarios when making decision:

#1 The valuable data you mention, may be lost easily in case there is fire/flood/earth quake. Is that OK? If so, local option works fine. If not, remote service may be better.

#2 If you need to have OS X's Time Machine (or similar Windows/*nix softwares) like fascility, Remote services are out of question.

#3 If you need "instant" transfers of large data files, Remote services are not the best bet

#4 If money is a major issue, and you would rather have your data close to you instead of on someone else's server, Remote is not a good option. In this case, I'd suggest that you buy two drobos. One is stored at your place, one at your buddy's (or someone else you trust) place. Your drobo will get backed up every night, and you can sync those backups with the other drobo at different location every week or so. This will somewhat help you cope the "earthquake" "fire" or "flood" kind of fears.

I hope this helped.

Both, of course. You do wear a belt and suspenders, don't you? There is no such thing as too much paranoia when it comes to protecting your data.
What happens if your house burns down? What happens if Amazon goes belly-up? It won't bother you a bit if you have DV copies in a Swiss bank safe deposit box.

Well since you are backing up home videos, you should probably go with and online service. It doesn't matter how many local copies of your data you have, if your house burns to the ground, they would all be gone. The only real problem with an online service is the monthy fee and having to upload all of that data over the Internet. It would also be a good idea to make a local copy for protection against system failure, otherwise you will have to redownload all of that data, but this would only be a conveince. What are the chances of loosing both copies at the same time? So, to sum up, an online service is probably your best bet. A local copy in addition to that is convenient, but not necessary as long as you always keep at least 2 copies somewhere.

I use Amazon S3 - pretty cost effecient if you backup under 10TB. if you compare annual cost of storing 10TB of data on Amazon S3 with the cost of external hard drive of the same size, i think you might want to consider the latter. I have my own freeware tool for Amazon S3 if you want to check it out http://cloudberrylab.com/

Why not use both? An external USB drive is fast, cheap and effective.

Ideally you would:
1) Have a backup locally on a NAS/Backup Device
2) Have a DVD/media backup
3) Backup online. Mozy is now owned by EMC which should help their reliability.

Remember to test Restoring you backup every now and again as unless you test the restore, you can't be sure the backups are correct!

I think that depends on how much and what kind of data you're storing, and how much it would affect your work if you were to lose data.

I think that the online services aren't worth the price for most consumer usage. I backup on external HDDs, on our local network, the most important files on DVDs, just in case.