Most of you guys may know the pebble watch, this watch receives push notifictions of you phone.
They are capable of receiving some notifications.
Try to check out feature Apple Notification Center Service (ANCS) https://developer.apple.com/library/ios/documentation/CoreBluetooth/Reference/AppleNotificationCenterServiceSpecification/Specification/Specification.html
It uses Bluetooth and talks through that to access all that data. It does not do anything fancy. However, all of that data is coming from the OS, not from applications on the device.
To answer your question - there is no way to get all of these notifications in your own app. You can use the CoreTelephony to access some of the data related to incoming calls and you could integrate Facebook and Twitter APIs to get that data, but you really would have to do it all yourself. There's no silver bullet here unfortunately!
Pebble uses Bluetooth to communicate with smartphone.Please find the answers for iPhone(iOS6) and relevant links.
Incoming Caller ID CoreTelephony Framewerk
http://developer.apple.com/library/ios/#documentation/NetworkingInternet/Reference/CoreTelephonyFrameworkReference/_index.html
You have to use PBAP to show the callerID on the bluetooth device and the device has to be MFi to communicate with iPhone app.
Email (Gmail or any IMAP email account) PushNotifications
SMS and iMessage PushNotifications with MAP
https://developer.apple.com/library/mac/#documentation/NetworkingInternet/Conceptual/RemoteNotificationsPG/ApplePushService/ApplePushService.html
Calendar Alerts EVentKit Framework
http://developer.apple.com/library/ios/#documentation/EventKit/Reference/EventKitFrameworkRef/_index.html
Facebook Messages Social Framework
Twitter Social Framework
http://developer.apple.com/library/ios/#documentation/Social/Reference/Social_Framework/_index.html#//apple_ref/doc/uid/TP40012233
Weather Alerts Any weather API from yahoo or MSN
Following code can be used to vibrate iPhone
Add AudioToolbox frame work import AudioServices.h
AudioServicesPlaySystemSound(kSystemSoundID_Vibrate);