Does time machine only work if you have an Apple wireless device?

just realized that you may be conflating time machine (software) with time capsule (hard drive in a wireless device) ... the time machine software will work if you plug a hard drive directly into your computer, but i'm not sure if it will work wirelessly if you don't have an airport extreme.
BTW, i plug my hard drive directly into my airport extreme and it works fine (although backups are inexplicably slow).

Your second answer is exactly what I'm dealing with. I cannot use the time capsule because it wants an Apple Wireless device. I guess I'll just buy a hard drive to use for backups, as you suggest. Thank you for explaining. I do think it's ridiculous of Apple to require their own wireless hardware, though!

ok, just to clarify ... Do you own a time capsule? it is a physical device that is about the size of a stack of CDs. If you do, the time capsule is also a wireless device and you need to use it as your router for it to work. You can get rid of your old router and just plug your internet connection directly into the time capsule ... it will work the same (or better) than your old router.

Time Machine is just the name of the back up software and it requires that you buy an external hard drive (or time capsule). otherwise, what would be the point of backing up?

No- I had no idea that a time capsule is a wireless/storage device. Interesting that the Apple Store employee didn't try to sell me one, as it would have been a natural "upsell". But I think I'll stick to the hard drive idea, since my current router supports several PC and Mac-based devices, and works great. Thanks for clarifying.