Screenshots (a.k.a. screen captures or screengrabs) are images of your screen that come in really useful when you need to show tech support what's going on with your phone or just to share your setup (e.g., at a homescreen-sharing site like My Color Screen. For Android phones and tablets, there are a few ways you can get that screengrab--from built-in methods to special apps on Google Play.
Android 4.0 (Ice Cream Sandwich): Hit the Power and Volume Down Button Simultaneously
Google introduced a screenshot-taking feature (which Apple's iOS already had) finally with Ice Cream Sandwich, Android 4.0. If you have Android 4.0 or later, you can take a screenshot of your device by pressing the power button and the volume down button at the same time. It might take a little bit of trial-and-error to master that simultaneous pressing. (I find that holding the buttons down for at least more than a second together avoids the annoying situation when your phone simply turns off the screen or lowers the volume.) You'll find the screenshots in your Gallery under a Screenshots folder.
Use Your Phone's Built-in Shortcuts
Some phones come with screengrabbing utilties built-in. Many Samsung devices, such as the Galaxy SII and Galaxy Note, for example, let you press the power and home button together to take a screenshot and place it in your Gallery. To find out if your phone has a screenshot tool, either check the manual or do a simple Google search for "[name of phone] take a screenshot]".
There might also be a device-specific app you can download to take screenshots and also do more with those images of your screen. For example, Screen Capture Shortcut Free works with some Samsung devices and lets you take captures after a delay or when you shake your phone. For other devices. search the Google Play Store for the name of your device and "screenshot," "screengrab," or "screen capture."
Install an App for Screenshots
If you don't have ICS or your phone doesn't have a built-in screenshot feature, you can try an Android app that does or doesn't require rooting your Android device. (Rooting gives you more control over it, so you can do things like tether your phone to serve as a modem for your laptop (without the onerous fees) or give a third-party app permission to take a picture of your Android phone's screen.)
If your device is rooted, you can use a custom ROM (e.g., Cyanogenmod) which has a "capture screenshot" option when you long-press the power button. An alternative is to use one of the many apps available that will let you take a screengrab. aScreenshot is a free one and AirDroid, which wirelessly manages your Android device, also lets you take screenshots of your Android device--wirelessly, even, through your computer's web browser.
Use the Android SDK
Finally, you can take screenshots of any Android device by installing the Android SDK from Google on to your computer. The Android SDK is a software development kit used by developers to create and test Android apps, but it's freely available to everyone. To use the Android SDK, you'll need the Java SE Development Kit, Android SDK, and possibly USB drivers for your device (found on the manufacturer's website). Then you plug in your phone, run the Dalvik Debug Monitor (DDMS.BAT file), and click on Device > Screen Capture... in the Dalvik Debug Monitor menu. It's kind of a clunky way to take screenshots, but if nothing else works or you have the Android SDK set up anyway, it's easy to use.