I'm not sure there's any one thing. When I think of New York City, which I would say is not only vibrant but nearly vibrating with energy, it's some combination of the following:

* A ton of different languages being spoken within earshot at any one time

* Lots of different smells at any one time, some delicious and certainly some much less so

* Lots of music where you do and do not expect it, all sorts of different genres sometimes complementing and sometimes juxtaposing the surroundings

* People physically moving and speaking faster

* The physical tightness of the city probably adds to all of the above

But most importantly, I think the above compose into an overall sense of serendipity. The phrase, "in the city, anything is possible" is frequently interpreted to mean, "you can achieve the highest version of what you want to achieve!" I think that's actually incorrect. It means exactly what it says: anything is possible. Relative to suburban or rural areas, you simply have no idea what is coming next minute-to-minute. That, I think, gives a sensation of constantly anticipating serendipity, which we round out to the word "vibrant."