I need to show the latest tweet of a specified user using the twitter stream API. I have moderate PHP, JSON and Javascript knowledge but not enough to figure out this twitter stream API on my own. Can someone explain to me what it would look like to "connect" to this twitter stream and display the latest tweet? Does something need to be done on the actual server, or can this all be done just through some json and php code in the website template?
You don't want to use the streaming API for this.
You'd want to fetch statuses/user_timeline and parse it. You could load it and parse it entirely via JavaScript, in fact, avoiding any need to do PHP work here.
If your set on using the Streaming API for whatever reason. What you could do is use something like PHP phirehose or to make it even easier as in dead simple easy. Look into and read the information posted about the 140 DEV Server by Adam Green.
His website is here: http://140dev.com/
and you can download the PHP framework/server from here: http://140dev.com/free-twitter-api-source-code-library/twitter-database-server/download/