Guys... this works

worked for me atleast :P
Okay, so I got rid of iCloud and I started receiving "Apple ID" sign in problem. Open an application and it ask for my Apple ID and Password and I fill up and nothing happens. Just like you guys have been facing.
Here's the SOLUTION:
***Settings > iTunes & App Store > Apple ID:
xxxxxx@xxx.com > View Apple ID (enter your password) >iTunes in the Cloud (just get rid of it)
Now you are all good to go. Have FUN
***EDIT***
Guys so I guess I was a little wrong about getting rid of iCloud or maybe iCloud does matter but never mind.
Even after getting rid of iCloud if you are still facing the same problem do the following:
*connect your iPod/iPhone/iPad to you PC/Mac and open iTunes
*Under the DEVICES header select you device
*navigate to Apps
*scroll below on the same page and *UNCHECK* automatically sync new apps
*delete the apps that were giving the APPLE ID sign in error(well that's what I did and later installed it after a restart of my phone)
*restart your phone(reinstall back the apps)
***NOTE***
I use iCloud only for Find My Phone