I've often wanted to show what I was telling, in a Stack Exchange question or answer (example). What would it take to make that super convenient? I imagine a third option in the Add Image dialog box, like so:
Is this something that could be added?
If you want screenshots of your desktop though - other pages, or other windows - that doesn't happen without plugins or external software.
Your browser page simply doesn't have access to the rest of your screen, only the stuff that's on that same page. This is a good thing! Websites shouldn't be able to arbitrarily access your screen anyway. It would be a pretty major security issue if one site could watch your entire screen whilst you dig around in your password vault program or type in your credit card number to another site. Pages are siloed off from mostly everything else and you should be glad about that.
However, there are browser plugins like Lightshot which can take screenshots of the browser window, as well as other parts of your screen. They can do this because they operate at the level of the browser software; they're not inside the page's silo. There's other software like Greenshot which you run on your PC, and Windows 7 onwards has Snipping Tool for cropped screen captures.
So if Stack Exchange wants to enable you officially to take screencaps of your screen, it's going to take making a plugin or program.
Might be better to just download Greenshot, save your screencaps and upload them.