Question: I am not sure whether this question is a good fit for this site, but if it is not, please let me know and I will take it down. 

If it is off-topic, some general info on where I can look for these answers would be greatly appreciated.

I am wondering how the data that determines how text is to be formatted is stored. For example, if I format some text in **Bold** and *Italic* in Microsoft Word, I can paste it into google docs and the formatting will stay the same, but If I paste it in here, it will loose its formatting. I am wondering if this is because google docs is aware of the formatting used by Word and it converts it, or if they use the same type of formatting and there is some set of standards.

Answer: 
Some applications, among which LibreOffice at Linux, use multiple clipboards if I'm not mistaken. 

The `clipboard` is the place where things are temporarily stored when pressing CTL+C, and of LibreOffice I know it saves it in different `clipboards`, in different formats. These formats vary from plain-text to rich and might even include images. 

There's also a function called `Paste special...` in LibreOffice, which allows you to choose how you want to paste the information in the clipboard. 

(what I'm saying might not be 100% correct - but I believe it is) I know you're using Microsoft Word as an example, so Widows might be a whole other world, but I hope it was of some help.
When you paste it here (TEXTAREA), the browser looks only at the plain text flavor of the clipboard.

When you paste it in google docs (["content editable"](https://developer.mozilla.org/en-US/docs/Web/Guide/HTML/Content_Editable)), the browser looks at the rich text flavor of the clipboard.