There's a few different things going on there:
if @controller.controller_name == "projects" || @controller.controller_name == "parts"
this gives the behaviour you want I'm assuming. The logic is pretty basic: return true if the controler name is either "projects" or "parts"
Another way of doing this is:
if ["projects", "parts", "something else..."].include? @controller.controller_name
That will check if the controller name is somewhere in the list.
Now for the other examples:
if @controller.controller_name == ("projects" || "parts")
This won't do what you want. It will evaluate ("projects" || "parts")
first (which will result in "projects"), and will then only check if the controller name is equal to that.
if @controller.controller_name == "projects" || "parts"
This one gets even wackier. This will always result in true. It will first check if the controller name is equal to "projects". If so, the statement evaluates to true. If not, it evaluates "parts" on it's own: which also evaluates to "true" in ruby (any non nil object is considered "true" for the purposes of boolean logic")