If you lose sound for keyboard clicks, games or other apps, email and other notifications, system sounds have probably been muted.
System sounds can be muted and controlled in two different ways: either by the side switch (located next to the volume up/down controls) or in the Recent Apps tray. The initial default is via the side switch.
Check the status of Settings > General > Use Side Switch to:. If this is set to Mute, the switch on the side of the iPad will mute system sounds.
If this is set to Lock rotation then the mute system sounds control is in the Recent Apps tray. Double tap the home button and, in the tray which appears at the bottom, swipe all the way to the right. The speaker icon is on the left. Is it is crossed-out tap on it and mute will be turned off and system sounds will be turned ON.
This Apple Support article explains in more detail how the side switch works.
As far as I can tell, 6.1 has changed it so that only Apple apps can play over the speakers. E.g. Videos/Itunes play fine oer the speakers. Games apps (e.g. Angry Birds Seasons HD, Pinball Arcade) play OK through headphones, but when you pull the headphone jack out, there is no sound. The volume up/down controls work fine, everything works as expected, headphones work fine just no sound from the speakers. I've tested on both my iPhone 4 and my iPad mini. I used to enjoy playing Pinball Arcade over the speakers (daily, for as long as I've owned the mini), but that no longer works for me.