How to pass an array within a query string?

I don't think there's a standard.
Each web environment provides its own 'standard' for such things. Besides, the url is usually too short for anything (256 bytes limit on some browsers). Of course longer arrays/data can be send with POST requests.

However, there are some methods:

1. There's a PHP way, which uses square brackets ([,]) in URL queries. For example a query such as ?array_name[]=item&array_name[]=item_2 has been said to work, despite being poorly documented, with PHP automatically converting it into an array. Source: https://stackoverflow.com/a/9547490/3787376

2. Object data-interchange formats (e.g. JSON - official website, PHP documentation) can also be used if they have methods of converting variables to and from strings as JSON does.
   Also an url-encoder (available for most programming languages) is required for HTTP get requests to encode the string data correctly.

Although the "square brackets method" is simple and works, it is limited to PHP and arrays.
If other types of variable such as classes or passing variables within query strings in a language other than PHP is required, the JSON method is recommended.

Example in PHP of JSON method (method 2):

$myarray = array(2, 46, 34, "dfg");
$serialized = json_encode($myarray)
$data = 'myarray=' . rawurlencode($serialized);
// Send to page via cURL, header() or other service.

Code for receiving page (PHP):

$myarray = json_decode($_GET["myarray"]); // Or $_POST["myarray"] if a post request.

A query string carries textual data so there is no option but to explode the array, encode it correctly and pass it in a representational format of your choice:

p1=value1&pN=valueN...
data=[value1,...,valueN]
data={p1:value1,...,pN:valueN}

and then decode it in your server side code.

Here's what I figured out:

Submitting multi-value form fields, i.e. submitting arrays through GET/POST vars, can be done several different ways, as a standard is not necessarily spelled out.

Three possible ways to send multi-value fields or arrays would be:

• ?cars[]=Saab&cars[]=Audi (Best way- PHP reads this into an array)
• ?cars=Saab&cars=Audi (Bad way- PHP will only register last value)
• ?cars=Saab,Audi (Haven't tried this)

Form Examples

On a form, multi-valued fields could take the form of a select box set to multiple:

<form> 
    <select multiple="multiple" name="cars[]"> 
        <option>Volvo</option> 
        <option>Saab</option> 
        <option>Mercedes</option> 
    </select>
</form>

(NOTE: In this case, it would be important to name the select control some_name[], so that the resulting request vars would be registered as an array by PHP)

... or as multiple hidden fields with the same name:

<input type="hidden" name="cars[]" value="Volvo">
<input type="hidden" name="cars[]" value="Saab">
<input type="hidden" name="cars[]" value="Mercedes">

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NOTE: Using field[] for multiple values is really poorly documented. I don't see any mention of it in the section on multi-valued keys in Query string - Wikipedia, or in the W3C docs dealing with multi-select inputs.

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UPDATE

As commenters have pointed out, this is very much framework-specific. Some examples:

Query string:

?list_a=1&list_a=2&list_a=3&list_b[]=1&list_b[]=2&list_b[]=3&list_c=1,2,3

Rails:

"list_a": "3", 
"list_b":[
    "1",
    "2",
    "3"
  ], 
"list_c": "1,2,3"

Angular:

 "list_a": [
    "1",
    "2",
    "3"
  ],
  "list_b[]": [
    "1",
    "2",
    "3"
  ],
  "list_c": "1,2,3"

(Angular discussion)

See comments for examples in node.js, Wordpress, ASP.net

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Maintaining order: One more thing to consider is that if you need to maintain the order of your items (i.e. array as an ordered list), you really only have one option, which is passing a delimited list of values, and explicitly converting it to an array yourself.