How do I use arrays in cURL POST requests

You are just creating your array incorrectly. You could use http_build_query:

$fields = array(
            'username' => "annonymous",
            'api_key' => urlencode("1234"),
            'images' => array(
                 urlencode(base64_encode('image1')),
                 urlencode(base64_encode('image2'))
            )
        );
$fields_string = http_build_query($fields);

So, the entire code that you could use would be:

<?php
//extract data from the post
extract($_POST);

//set POST variables
$url="http://api.example.com/api";
$fields = array(
            'username' => "annonymous",
            'api_key' => urlencode("1234"),
            'images' => array(
                 urlencode(base64_encode('image1')),
                 urlencode(base64_encode('image2'))
            )
        );

//url-ify the data for the POST
$fields_string = http_build_query($fields);

//open connection
$ch = curl_init();

//set the url, number of POST vars, POST data
curl_setopt($ch,CURLOPT_URL, $url);
curl_setopt($ch,CURLOPT_POST, 1);
curl_setopt($ch,CURLOPT_POSTFIELDS, $fields_string);

//execute post
$result = curl_exec($ch);
echo $result;

//close connection
curl_close($ch);
?>