Years ago, text could only be copied in Windows between the programs as a plain text. That is, only the content, without the format. Today the situation has
improved changed and in many programs now when you copy and paste the text, you do so together with the format, whatever it may be.
In my opinion, this is a really poor approach from the user experience perspective. You should use styles to control the look of your text, so the text properties should not be connected to its content. One should really move around the text, not the look. (Word actually does it pretty neatly, by moving the style rather than the look of the text. The problem however is still there if you copy & paste the text from Word to, say, Excel.)
Another problem with moving the text properties around is that sometimes it is really not what you want. Have you ever tried to copy the content of a web page and paste it into Word? You know what happens. Word tries to keep the whole structure of the web page and transfer it to your document. The problem is that web pages are formatted inherently differently than the paper documents. Adding the structure of a web page to a text document means adding numerous redundant elements, such as tables, to your document. Besides, you just wanted the content, the text.
In Google Documents with Windows + Chrome, you can now press Ctrl+Shift+V. For other OS and browsers, see here.
In MS Office, use the Paste Special command. And if you do not like using it all the time, the PureText program comes to help. It is beautifully simple, tiny, and does not require installation. You can assign a new combination for pasting text-only (such as the default Windows+V or Google-like Ctrl+Shift+V) or overrule the standard Ctrl+V. The only two drawbacks are that the program needs admin rights to be run and is not as fast as Windows own clipboard.