Culture is about unspoken expectations and what people do that they're not specifically told to do.
When people finish a project 10 minutes before they planned to grab lunch, do they leave for lunch early? Do more of their own work? Sit down and help out a colleague? Chat about Game of Thrones by the water cooler? Play ping pong?
Speaking of lunch, do people eat at their desks? Together in the kitchen or a conference room? Alone outside the office? Together outside the office?
At the end of a hard day, what do people do? Stick in their chairs and surf Facebook so the boss thinks they're putting in lots of hours? Go home to their kids? Get into nerf gun wars? Play video games together? Have a beer together in the office? Go out to a bar?
How competitive is your culture? Do people default to collaborative modes instead? Where there is competition, is it friendly and constructive or vicious and inclined to sabotage?
How much autonomy do people really have? How much hierarchy? How much process and bureaucracy? What do you look for on a resume and in an interview? How do your managers identify good work and how do they recognize and reward it? Do certain roles get more appreciation than others? Do people speak out loud in your open office, or whisper out of consideration for the people around them or take their discussions to a conference room? Do people actually take vacations or does that unlimited vacation policy just make them afraid because they don't know how much they're expected to take?
These questions and a thousand more define the cultural differences between companies. They're the reason I might consider working at a large company (Google? IBM?) but certainly wouldn't work at Amazon or Goldman Sachs. They're why I like the small startup-minded company I'm at now but not the culture at most Kool-aid drinking startups. They'll affect what sorts of people you retain and what sort of results you get out of them. They affect morale and (via morale and burnout, among other things) productivity.
Now here's the crux, and this is very important:
Culture is introduced from above AND from below.
From above, you're responsible for what personalities you hire and what skills and personal values they bring. You're responsible for what work behaviors you reward. You set the example for how rigid hierarchy is, for how much vacation people take, and for whether people are expected to spend 7 hours in the office or 11.
But you can't force-feed culture either. Culture is set from below by the personalities and personal values of the people you hire. How well they get along. What they like to do when they have 10 minutes free or after work.
If your employees are more into knitting or WoW, buying a ping pong table won't suddenly make them ping pong players. Taking everyone to a bar and forcing them to socialize won't make them friends, but it might make some people resentful.
All you can do is avoid hiring (and quickly fire) poisonous personalities, and offer encouragement and support (in the form of money or time) to activities that people already express interest in as a group and that you see as constructive.