This is a great question. I have a bunch of ideas about it, but I'm not even sure that these correctly reflect the actual criteria that go into my subjective impressions of a bike. This is just what my conscious mind tells me I care about. I think I have these in more or less the order of importance to me.
1. Fit -- I don't know when this happened but I have become princess-and-the-pea sensitive to bike fit. I've got a set of measurements I use to set up my bikes and I'm at the point where I can actually feel when my saddle is 1/8" too low and when I get home measurement confirms it. I try to get every bike set up exactly the same way, but with limited range of fore-aft adjustment on the saddle rails and 10mm increments in stem length I can't get every bike precisely the same. I have one bike that within three pedals strokes of leaving the driveway gives me that "Ahhh..." feeling like you get when sinking into a comfy recliner. Given how immediately that feeling appears, it can only be the perfect fit of the bike. I'm constantly measuring my other bikes to see if there's something I can change to make them more like the magic Lemon D.
2. Silence -- I'm nearly as OCD about my bikes being silent as I am about the fit. I do things like putting a thin strip of rubber between my fenders and their mounting bracket because a little bit of rattle drives me crazy. If anything clicks, creaks or squeaks I am hunting it down. If I could find tires that didn't make road noise, I'd probably buy them. Objectively, most noises probably don't have any effect on the performance of a bike, but psychologically I think it's a huge factor in my impression of how the bike "feels".
3. Responsiveness -- I feel like this is still a pretty vague and ambiguous term. A lot of different things can ruin the responsiveness of a bike. For me to think a bike really rides nice they all have to come together to give me that feeling that I somehow am faster than usual when riding that bike. I guess this is part of what marketing types are trying to get at when they talk about a bike being stiff. I don't personally associate any lack of responsiveness with flex, but maybe that's what it is. I think of it as a sluggish feeling at the pedals. With some bikes when I spin the pedals it feels like I have a flat tire or the brake is dragging or I'm riding through mud or riding into a headwind or all of these. With other bikes, I move my legs, the pedals spin and the bike goes forward; it doesn't "feel" like anything is resisting me. The things that I believe contribute to this are (probably in this order): (a) gearing, (b) tire construction/suppleness, (c) tire inflation, (d) drivetrain cleanliness, (e) bike geometry, (f) hub and bottom bracket bearing smoothness, (g) tire and wheel weight, (h) tubing material and shape.
4. Cornering -- Once a bike fits, is quiet and is responsive, the next thing that separates one bike from another for me is the way it corners. Until about five years ago I rode mostly cyclocross and mountain bikes and this wasn't even something I considered. Then one day I decided to buy a road bike (a LeMond Nevada City) and the first time I rode it I was just going through a residential neighborhood and making a 90 degree turn that I didn't need to slow down for and it was like "Wow! What was that?" Since then I've bought more and more road bikes and even in my cyclocross bikes this is something I focus on. I guess this is all about bike geometry.
I'm not even sure what I prefer -- different days I want different feeling for cornering. That's part of how I justify my N+1 habit these days. My LeMond dives into corners and feels like it's pulling me through, like the bike wants to go faster. My Pinarello Gran Turismo feels like it's somehow straightening out the road, it just really flows through corners. My Trek kind of dives like the LeMond but it pops back up quicker and feels eager to go straight again. All of these are fun, but they're all different.
On the flip side, I've had bikes that felt like they were trying to talk me out of going around turns. Navigating a windy road was like reading "The Monster at the End of this Book" -- "Oh no, another turn...wouldn't your really rather keep going straight?" I guess that's what people who prefer stability are looking for. I had a Surly Long Haul Trucker that was like that and most of the time I didn't like it, but when I actually loaded it down and went touring, I was like "Oh, OK. I get it now."
5. Shifting and braking -- This would be higher on the list, but I'm less likely to blame bad shifting or braking on the bike itself. That's either my fault or the fault of cheap components. Either way, it's generally fixable. Of course, nothing can ruin a ride for me like bad shifting (except maybe bad fit or constant creaking).
6. Buzz absorption -- This also could be much higher on the list, possibly as high as number 3. The reason I put it here is that I hardly ever find big differences in this area between bikes that pass the first two criteria. A bike that fits comfortably and doesn't make noise is probably absorbing road buzz pretty well. That is until I get to a road with chip seal, and then my experience is that there's nothing for it except fatter tires.