Unfortunately, not every phone is so simple. Let's start with earlier Android versions. First, let's talk about run-of-the-mill Android phones, which are "unrooted." (We'll discuss this in detail later, but this is your regular, out-of-the-box Android phone.)
There's no simple button pressing for early Android users. What you can do is go to the Android Web site and download the Android SDK. SDK is a development tool that's downloaded to your computer -- note that: not your phone. Once it's installed, plug your phone in through a USB cord and go to the "tools" section. From there, click on the "ddms" icon. Find your phone's name and then go to the "device" icon. Click on it and you'll see "screen capture." You'll see an image of your phone; find whatever you want to capture (on your phone) and simply press save to take a picture. Yup, not exactly simple.
If you have a rooted phone, you might have better luck. Rooted means the phone can perform all sorts of tasks that the Android operating system doesn't want it to; for instance, running apps from the Apple Store. So you can run apps that will capture the screen, but keep in mind that rooting your phone will void your warranty and isn't exactly encouraged.
You think that was tough? If you have a Windows phone, you might be even more disappointed. Because with the current Windows 7 device, you literally can't capture the screen.
Ok, maybe not entirely. If you have a developer unlock your phone (basically meaning you can test apps without having to go through the Windows Marketplace), you can gain access to a few apps that will help you capture your screenshots. However, some of them do require a PC connection.