What does it mean when a PHP function name begins with an underscore?
for example: __construct()
I know what the construct means but I've seen other places where the function begins with an underscore, or a double underscore and I'm just not sure of the significance.
It means that PHP calls it implicitly.
It's called a 'Magic Method'
Also, it's two underscores, not one.
Learn more here: PHP Magic Methods
In PHP, functions start with two underscores usually have special meanings. From the manual:
For example __construct() is a special method which is called automatically while initializing an object.
Se also: http://php.net/manual/en/language.oop5.magic.php
As noted in other answers, the double underscore is used for "magic" methods. The idea is that the user would never intentionally use two underscores for a method name, so there is little risk of collision. The reason it isn't a single underscore, I believe, is that the single underscore was a popular convention for private methods in the bad(der) old days, before the 'private' keyword came along to enforce OO privateness.
In codeigniter, if you start a method in a controller with an underscore, the router will know not to allow it to be accessed from the url.
There's also a common naming "coding style" which uses one initial underscore to indicate methods or properties are private/protected. I think it's pretty widespread.