If you enabled the Today View when you set up push notifications, then Today is the first option when you open Notification Center. It's designed to give you all the information you need to be prepared for your day.
First you'll see the day of the week and month and date, followed by a weather forecast for your area (this assumes that you've got GPS turned on to allow your device to figure out where you are).
One really neat feature of this screen is that it can tell you how long it will take you to drive to work or school (in the screenshot, notice the section that reads "Right now, it would take you about 1 hour 2 minutes to drive to Boston.").
In order to get that kind of information, you need to have Frequent Locations turned on the Location Services settings. This setting lets your phone observe your behavior and learn your schedule. For instance, my phone has noticed that I go to Boston every morning for work and is smart enough to know that I'm probably going to do that today and might want information about the commute.
The Day's Events
After that section, you'll see your calendar of events for the day. Events will only display if you're using the built-in iOS Calendar app. Third-party calendar apps are not allowed to show events there as of this writing. If you want to learn more about an event, tap it and you'll be taken to the Calendar app.
You'll also get a preview of tomorrow, with a listing of how many events you have and when the first one is scheduled to begin.
Once you've got a handle on the day, swipe right to left or tap the All menu at the top of the screen to see more notifications.