Due to the complexity of using the twitter API you want a PC (webserver, probably) to handle scannign twitter to find a new and relevant tweet, then open a socket to the arduino and send the command to puff up. You'll have to google about the twitter API, but it requires the use of oauth these days so older code won't work.
Have a look at the chat server example - that'll give you a good start on setting the arduino as a server. Have it wait for a machine to connect and send a particular command.
edit: It might be easier to have the arduino ask the server if there are any new tweets. So your arduino would visit http://YourServer.com/something.php and would be returned a Y or N for whether it should puff up or not.