Array is a data structure with certain properties, like it has length (the number of elements it contains), it can tell the position of every element and it promises that the time required to access an element, given you know it's position is "immediate".
Quite by chance
there is a class in AS3 called Array - strictly speaking it doesn't exactly satisfy the description of what arrays should be, but it is close enough.
In AS3 you create arrays by either using a literal initializer of the form:
[element_0, element_1, ... element_i] - which creates an element of the length i containing elements element_0 through element_i.
Alternatively, you can use new Array(i) to create an empty array of i length.
There are less often used ways of creating arrays: new Array(element_0, element_1, ... element_i); is essentially the same as the first one described (the details of the difference aren't really important for now). Or, you could even write Array() - and it, too, will create a new array - but this is confusing syntax-wise, and you probably should avoid doing it.
When you call addChild() or removeChild() you tell the display object container, which owns these methods to add or remove children. Eventually, the container may choose to store children in an array, but, in reality this is not what happens, besides, you don't really have a way to know how exactly the children are stored, and, ideally, it shouldn't even concern you (unless in the corner cases, like that when removing many children at once - it's useful to know that removing from the head is faster then from the tail).
Re' your other question... text files so far have been the best medium for writing and storing the code. Some modern code editors try to change that, transforming it into a sort of MS Excel tables... which I personally see as a waste and a total misconception, but wouldn't like to start a flame war.
Writing code is a different kind of activity (different from graphic design) - it is in it's nature more similar to the work of a writer or a scientist - in both cases, it's natural to have it written or typed. It's rather uncommon to do it in another way. Flash is quite particular in this sense because it tries to combine the process of writing the code with the visual graphic tool. Probably, you could take advantage of both worlds, programming and designing, but most commonly you specialize on one side of it.
However, if you intend to be a programmer, AS3 or anything else, then writing your code in text files (in the context of Flash) is the way to go, because it opens the door to better code editors, but not only that, it also trains you to work in the environment typical for a programmer - which is often suited for more then a single programming language. Things like source control (versioning) systems, organizers of all sorts, code statistics and many other useful programming tools rely on that the text of the program is easily accessible as text files.