Attempting a semi-generic answer (which won't match everyone's experience, to be sure, but hopefully will be more generic than some, allowing for less duplication of concepts -- if someone wants to turn this answer into a community wiki, please feel free):
The camera that made me fall in love with photography was the all-mechanical SLR (except for the electronic light meter, which you could use the camera without having batteries for) that my parents let me use. In my particular case, it was a Nikon FM, but I think the same story could apply with any number of cameras (e.g. another poster's Olympus OM-1). The things that made the difference for me were:
It had a macro lens. Being able to take a photo of something and have it appear life-size (actually slightly bigger) on the film was a wonder to me. From pennies to human eyes to rainbows from a crystal window hanging, falling across the patterns of denim, this somehow got me excited about things (I'd later get into much more interesting macro subject matter than some of those).
All mechanical. There's a certain sound about things, as the other poster talks about, but also it allows for one to explore the workings -- to open up the camera and see what happens as you slowly re-cock the shutter, for example, or to find the little lever that interfaces between the camera and lens to cause the aperture to close when you fire the shutter or hold the DOF-preview button. Not to mention the fact that it just sort of Always Works -- worst case, you stop having a light meter.
It had interchangeable lenses. In my case, besides the 55mm/2.8 macro, there was also a 70-200 zoom lens. I've always liked telephoto, somehow, so having this was cool. And having the old-school style where the outer barrel moved, and had the flaring DOF-guide, was just somehow satisfying. (If anyone doesn't know what I'm talking about, look at the lens in this video-explanation of DOF; he describes them about 1:45 in.)
It was an SLR -- I was actually looking through the lens that the picture would be taken through. This was magical to me, somehow -- and certainly an improvement over the alternative for doing things like macro.
It was what my mom was using -- and always nice, as a kid, to emulate mom and dad, right? :)
I could do long exposures with it. I think I'd used the camera before I really discovered the wonders of this, but BULB mode with a locking cable-release allowed me to take pictures that were unlike the world I saw with my naked eye, which was really neat to me.
Beyond just long exposures, it also just let me have control of exposure, and in particular exposure time. When I learned that to get a good photo of a TV (classic CRT, of course, and NTSC in my case), you wanted a shutter speed of 1/30 of a second. If you used a faster shutter speed, only part of the frame would be lit up. Wow. Only part of the frame is lit up at one time?? And that's not just something that someone has told to me in the abstract, but it's something I can actually take a photo to prove? That sure got me excited. And then I turned it around, and for a cell-animation of Snoopy, I ended up with a nice still image of the dog house, with four dancing snoopy poses ghosted on top of each other. How cool is that?
The optical split-screen focusing screen, with a ring around that of (not that I knew the name at the time) micro-prism indication... having a way to really know when things were in focus: very handy. But also fascinating -- "hey, weird -- if I move my eye around, half this inner circle goes dark. What's going on? That's weird, but cool!"
I'm sure there were other things, too, that I'm forgetting to mention.
The bottom line, for me, is that this was the camera that got me excited about photography. As other answers have said, getting into the darkroom later was a very important step, as well. But really, I was already hooked, thanks to this camera.
I'm even lucky enough to have had my mom give it to me, years later, when she got a DSLR. As with other posters, I feel guilty at times for leaving it on a shelf. But then, sometimes, I take it off the shelf, and shoot with it. And it's still a joy to me to do so. :)
Hopefully others can relate to a similar experience -- again, feel free to make this a community wiki and add your own bullet points, or just add them in the comments.
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