Taking a screenshot of your Android phone's screen can be a bit confusing, since the process isn't the same on every device. Here are the different ways to take a screenshot on Android.
If You're Running Ice Cream Sandwich (4.0) and Above
If you have a shiny new phone with Ice Cream Sandwich or above, screenshots are built right into your phone! Just press the Volume Down and Power buttons at the same time, hold them for a second, and your phone will take a screenshot. It'll show up in your Gallery app for you to share with whomever you wish!
If You're Running Android 2.3 and Below
Unfortunately, Android 2.3 and below doesn't have screenshotting built-in. However, some devices (like many Samsung phones) do have these features, but it varies from phone to phone. for example, on many Samsung phones, you can press the Home and Power buttons at the same time to take a screenshot. Google around to see if your own phone has a built-in shortcut.
If your phone doesn't have a shortcut built-in, you'll need to use an app like No Root Screenshot It. You'll need to install it on your phone and install its free companion app on your computer, which will enable screenshots. You'll have to plug your phone back into your computer every time you reboot it, but you'll be able to take screenshots directly from your phone which is great.
If You're Rooted and Running Android 2.3 or Below
If you've rooted your phone, you have a few options. You can install an app like Screenshot UX, which will allow you to take screenshots through a variety of shortcuts, or you could flash a custom ROM that has screenshotting built-in (like CyanogenMod).
There are more screenshot methods than we could possibly list here, but these are some of the easiest (if you aren't rooted but know how to use ADB, you can get a screenshot from the command line, for example). Check your phone and see what it supports out-of-the-box.