What is catcomplete?
catcomplete is a property that appears on jQuery objects. It is not part of core jQuery so there is presumably some other script on the page that is adding it.
How does jQuery’s autocomplete call it?
I can’t find catcomplete mentioned anywhere in the documentation for autocomplete, so it probably doesn’t. It just modifies the CSS of elements in the page with class names that suggest they are normally influenced by autocomplete.
What are minLength, source, open and select?
Properties of the object that gets passed as an argument to the catcomplete complete function. You should read the catcomplete documentation to find out what they mean.
Who decided their name?
The author(s) of catcomplete
Which part of the code knows how to use them?
The catcomplete script
Are they a part of standard jQuery?
No.
Is catcomplete a function?
Yes, it has (…) after it to call it and does not (presumably) throw an exception.
What is
$("#searchTextBox").Example({})?doing? Am I defining an array and passing it to a function called Example?
No. You are creating an object. An array would be [].
Or am I defining a function called Example which initializes an anonymous Array?
No. That would be:
function Example() {
var foo = [];
}