Is it possible to use Time Machine to back up to an external hard drive connected to a Mac on same wifi network?
Yes. I've been doing this since Snow Leopard. All my MacBook Pros in the house do their Time Machine backups to an external USB drive that's hanging off an iMac running the standard OS X install (i.e. not Server). At the time of writing the iMac has been upgraded to Lion as has one of the MacBook Pros.
First, configure your target machine (the one with the USB drive attached):
Now you need to configure your source machine (the one you want to keep it's Time Machine volume on this drive):
You're now all set up to use this remote drive as your Time Machine backup location. When Time Machine runs you'll see this drive appear as a mount on your system, and when it's done the mount will go away.
If you browse this drive from the target machine you'll see sparsebundle files, one for each machine that's doing a Time Machine backup to this drive, on the disk. These are the actual "disks" that Time Machine is using on each machine to keep the incrementals. If you click on one, it should mount and you should be able to browse it like normal.
If you have a complete disaster and need to do a fresh OS X install and recover from Time Machine you can point the recovery process at the remote drive and it will ask you to pick a sparsebundle from the drive to recover from. Or, and this is what I recommend, you can unattach the drive from your target and attach it via USB to your machine and do the restore over USB which is much faster.
I can confirm that complete restores from Time Machine backups kept in this manner work well. I've had to do two of them in the past few years on machines that keep their backups this way and both went smashingly well. Browsing Time Machine history in the Time Machine viewer can be a bit laggy, but it does work.
If the Mac is running OS X Server (In 10.7, not sure if this is available for 10.6) then you can turn on Time Machine Server, which does exactly what you need. There's no need to run any of the other server tools if you don't want. this is the only officially supported way to do this that I know of.
If your backups are really important and you need a much more stable system for doing backups either remotely, wirelessly, online or an external drive, then I describe the best way how to do this below.
A hard drive used for Time Machine backups requires an HFS+ journaled partition which gets corrupted and unmountable very easily. Your signal getting cut off, a cable getting detached, the power going off, or your computer crashing during a backup is all it could take to have big problems. Often Disk Utility cannot repair the disk and you lose all your stored data.
This is why using Time Machine over wireless is risky (time capsules are expensive but do seem to be more reliable); be really sure that you have a good, reliable WiFi signal and set up your router to use a channel different than any other WiFi sources. (The free program WiFi Scanner is great for analyzing WiFi interference.)
My backup hard drive had a habit of mysteriously disconnecting without properly ejecting numerous times and I have lost files; most of my video, graphics, and photos which take up more storage then my laptop can hold.
I do video editing and design and for my situation I need something more reliable and flexible. It's worth it to me to spend a little extra money on this software because losing data had cost me in more ways than one. Here is an excerpt from their website:
I use xTwin with an external USB teradrive. HFS+ Journaled formatting is so easy to corrupt and lose your data with but Time Machine requires it. So far the best solution I believe is to use xTwin with NFTS formatted drive. I learned that NFTS is much more stable and a lot easier to do data recovery with. Disk Utility does not come standard with this format, but you can get software such as Paragon which adds this option into disk utility. It is much more important to me to have reliable backups than versions and the cool time machine way of accessing older files.
I will stick with this until Apple proves they have made a more stable filing system for backups.
Not possible that I know.
Only if connected to AEBS (Airport Extreme Base Station) or server as far as I know