Does Word check spelling/grammar on Microsoft's servers or on my computer?
No it doesn't.
When you install Microsoft Office, it will install a dictionary of your main language - however, this isn't much more than a list of words. You can also install additional languages later. You do not need an internet connection for this.
In addition, in any Office program, you can hold down Alt and click on any word to perform research on it - this includes translating, dictionary (with full definition) and thesaurus. All of these research features require an internet connection.
(this information is current as of Word 2003)
All spellchecking is local. The spellcheck data is stored in the file
C:\Documents and Settings\<user>\Application Data\Microsoft\Proof. (WinXP path)
You can add, edit, and remove dictionaries used in Word; go to
Tools -> Options..., select the tab
Spelling & Grammar and click on
Here you can check what the file is on your system and it's location