Is there a built-in mechanism to take a screenshot on my Samsung Galaxy Tab 10.1? Or is there an app I can download to do so (without rooting the device)?
Unfortunately, it does not appear that straight Honeycomb (Xoom, Samsung 10.1, etc) devices have the ability to take a screenshot without some help.
If you have the SDK downloaded you are able to plugin your device and take screenshots using your computer. However, that sounds really inconvenient. Instructions can be found here under "Debugging Tips" http://developer.android.com/guide/developing/debugging/index.html
There was also an independent app released on XDA for Honeycomb devices called Screenshot ER. One caveat, it does require root. http://www.xda-developers.com/android/take-screenshots-of-your-honeycomb-device-with-screenshot-er/
Hold the Back button and tap the Home button (or, if that doesn't work, tap both at the same time). This is part of Samsung's TouchWiz interface, it doesn't work on other devices.
On most other devices, the only way to take a screenshot without rooting is to use the Android SDK. You would connect your phone to your PC via USB, run
ddms, select your phone, and hit Ctrl+S.