Checks spelling on entire page as requested (not only in text fields)

Simply find a text field and start typing to activate it.
Features
- Clearly shows the typos/errors on the page (underlines them with a a wavy red line)
- Works in Chrome (surprise!)
- Allows adding of words to custom dictionary (just right click)
However, it is not open-source. I believe a spell-check is in the works in Chromium however, which is open source. You might want to check that out.
If you happen to be disatisfied with the native spell-check for some reason, the following alternatives are available
This is a spell checker extension for Chrome.
It's easy to use and fast.
This extension supports 12 languages for spell check. It's a tool to check spellings of words and come up with suggestions for misspelled words.

The free Spell Checker & Grammar Checker by Grammarly helps you write clearly and confidently on Gmail, Facebook, Twitter, Linkedin, Tumblr, and nearly anywhere else on the Web. With Grammarly, you can always be sure that your words are mistake-free and say what you mean.

To spell-check the entire page
Chrome doesn't officially support this and I don't know of any extensions that do this. Instead, you could copy-paste the contents into a text box, or use this script:
javascript:document.body.contentEditable='true'; document.designMode='on'; void 0
Run that on any page you want to spell-check. You can even add it as a bookmarklet if you like for easy access (before & after pics below).

edit page
JS bookmarklet. Huey already posted it in his answer – Tymric Jun 4 '15 at 20:39