Very unusual indeed! When you place your finger on the app it should expand a little and then you can move it....don't know why it's not working for you. Do these steps and try moving it between each step.
Close all your apps completely. From the home screen do the four finger swipe up or double press the home button to reveal your multi task bar. These are all the apps that you have opened at one time. Long press the app until it jiggles and a red minus sign will appear above it. Press the minus sign, this will close the app, not delete it, now press the home button to return to your screen. It's also a good idea to close all these apps, as it helps the iPad run smoothly.
Restart - Press and hold the power off button, the one at the top right. You will see a slide appear to power off, slide this and the screen will go dark and a Whirly will appear. When the Whirly disappears your iPad is now off. Press the power button again until the Apple logo appears, this will take a few seconds and your iPad will restart. You should restart your iPad occasionally also as it will keep it running well. Now see if it is operating.
Reset - this reset will not change anything on your iPad and is akin to rebooting a PC. Hold the power off button and the home button simultaneously, keep holding them in and you will see the power off slide appear and disappear, wait until you see the Apple logo appear before releasing the buttons. It will take a little longer for your iPad to start.
Let us know how you go.
iPad 2 3G 64gb