An array is a list of items. The items in the array could be numbers, text, objects, or even other arrays. Don't try to make it more complicated than it is and just think of them as a list.
The more important question is...
Why do we use arrays?
Arrays are useful for grouping common lists of data into one variable. Even more valuable though are all the functions built into PHP to help manage lists of data.
For example, let's say I have an array of store names. I could alphabetically sort them by using the function: sort(). I could reverse sort them with rsort(), or I could even randomize the order of the array with shuffle().
There are many other functions specifically made for arrays, but I'm just giving a higher level explanation here.
Iteration is another reason to use arrays. When we have lists of data, we will often want to iterate through each one to either print to screen or maybe test for values. PHP offers a foreach() construct for the purpose of iterating through each item in an array.
It is the use of arrays and array specific functions that help us manage common data and saves us the hassle of writing and maintaining very difficult to manage code.
When should I use arrays?
Use arrays whenever you have a list of data. It can be static data that does not change such as the 50 states in the US, or it can be dynamic data that may load from a database such as the products of a store.
Any time you find yourself hard coding values in a drop down box, select box, or multiple radio buttons, you should ask yourself 'can i put these in an array?'
It will end up making your project much easier to code and even easier to maintain in the future.