Question: On iOS devices you can take a screenshot by pressing the home & sleep buttons at the same time. Is there a similar way to take a screenshot with Windows Phone?

Answer: 
In Windows Phone 8 you are able to save a screenshot by holding both the power button (often on the side of the phone, between the volume buttons and the camera button) and the home button (the middle button under the screen, with the Windows logo on it) for some time. Such screenshots are saved in the Screenshot album of the Picture Hub.

And just to be complete: in Windows Phone 8.1 Preview this combination was changed to holding both the volume up and the power button. If you use the older Windows Phone 8.0 key combination, you get shown a message box that tells you of this change.

For normal users (no developers) in Windows Phone 7.5 and earlier there is *no* way to make a screenshot of your phone except using another camera(phone) to do it. I think this is still the case in the Windows Phone 7.8 update that brings the smaller home screen tiles from Windows Phone 8.
Windows Phone 8
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Press and hold Start button and Power icon at the same time to take a picture of your screen.

Windows Phone 8.1
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Press the Power icon and Volume Up buttons at the same time to take a picture of your screen.

Windows Phone 7, 7.5
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It do not provide a way to take screenshots without unlocking the phone with developer tools.

*The screenshot will then be stored into a "Screenshots" album in the photos.*