It used to be that by typing a name, such as fatwallet, into the address bar it would open fatwallet. And it still does for FW, but if I type other sites such as "homedepot" it'll open a google search page.

Some sites open in google, others link to the site. This change seemed to happen with an update to 3.5.3

What setting do I change so it always opens the website without having to enter .com or .net or whatever?

it searches in your history and bookmarks. Make sure it is selected by going to tools-> options and then the privacy tab

there is a dropdown that says Location bar

when using the location bar suggest and you can choose history and bookmarks, history, bookmarks or none.

be warned if you have a large browsing history and a ton of bookmarks then it can cause a long time to open a new tab.

Many programs, even ones as popular as Realplayer, will let you accidentally install things like the "ask toolbar" if you forget to uncheck a box. If you install anything like this, even if you uninstall it, it can leave Firefox settings changed that will screw up auto-url-correction. If upgrading to 3.5.4 does not fix it, examine these settings (and possibly reset them to the default by right clicking), and make sure you don't have any toolbars installed in Firefox.
http://kb.mozillazine.org/Keyword.enabled
http://kb.mozillazine.org/Keyword.URL
These settings are in about:config, explained here.
http://kb.mozillazine.org/About:config