How can I use an app purchased from the Market on a Kindle Fire? I bought "Cut the Rope" on my Nexus One cellphone from the Android Market. When I open this game on my Kindle Fire it asks me to buy it but I expected it to be "Purchased".

How can I add my Google account to Kindle Fire's Amazon appstore account list?
The Amazon Kindle Store and the Google Android Market are different stores.  Apps you buy in one store are not honored in the other store.

You can install the Amazon Store on your Nexus One pretty easily, so you'll be able to use Apps you buy on your Fire on your Nexus One.  But going the other way is more trouble (it currently requires you've rooted your Kindle Fire).  See How can I get the Android Market installed on my Kindle Fire?