My criteria for what constitutes an awesome bike is highly variable. This is largely because I take a "Tool for the Job" approach to cycling, where no one bike is ideal for all tasks, though certainly some bikes are more ideal for most tasks than other bikes which are excellent at one thing, but suck at anything else.
There are no better examples of the two sides of the coin than a cyclocross bike and the downhill full-suspension mountain bike. For reference, I will choose two bikes I consider to be fairly representative of each type of bike. I'm going to go ahead and admit a huge personal bias towards smaller, quirkier manufacturers whose work is primarily in steel.
So check it. This is the Soma Double Cross:
This picture comes from the website for Soma Fabrications. However, when I googled this bike, most of the pictures of it came up tricked out as a commuter. For those not totally immersed in bike dorkdom, cyclocross is a sport that is basically centered around taking roadbikes off-road. This has created bikes which are fairly fast, handle very well in a variety of conditions, and tend to be tough as nails. They are versatile by their very nature, and as such, it's a bike you can ride for road rides, can do some trail on, and does just as well getting you to work. A lot of manufacturers have noticed this and include a lot of commute friendly features like braze-ons for racks and fenders and all the other goodies people like to put on their rides.
To contrast, this is a full suspension mountain bike, the Cannondale Claymore:
This bike is an amazing over-mountain bike. It will roll over just about anything going down a hill and you can do some incredible rides on it. But here's the thing - even going uphill on this bike is a bear of a task, due to the weight of the suspensions which make hillbombing at questionably intelligent speeds possible. If you tried to ride it on a normal trail with some people riding hardtail 29ers or god forbid, on the street with a few people riding hybrids, it would feel about like dragging a ball and chain up a hill. A squishy ball and chain.
The point I'm getting at here, is when someone asks me what constitutes a "good bike" or "the best bike", the first thing to ask is, "Well, what are you going to use it for?" because there is no one right answer. A good transport bike has a very different criteria from a road racing bike or a full suspension downhill bike.
For my money, versatility and practicality always win. Maybe I'm insufferably boring (not to mention preachy and arrogant), but if I'm blowing the equivalent of a month's worth of life on a bicycle, I like to be able to use it for more than just buzzing around looking cool. I also like using it for buzzing around looking cool while crossing things off my to-do list. Even if all that's on there is "Go to the pub for beerskis".