HTML

<form action="include/dbh.inc.php" method="post">
    <table id="table">
        <tr>
                <td>
                    <div class="white">
                        <input type="radio" name="first" value="white" onclick="setBgColor(this)" checked>
                        </div>
                    <div class="red">
                        <input type="radio" name="first" value="red" onclick="setBgColor(this)">
                        </div>
                    <div class="orange">
                        <input type="radio" name="first" value="orange" onclick="setBgColor(this)">
                    </div>
                    <div class="green">
                        <input type="radio" name="first" value="green" onclick="setBgColor(this)">
                    </div>
                <br>
                TEXT TEXT TEXT TEXT TEXT TEXT
        </td>

  PHP

<?php

// Här är din check på ifall "first" finns.
// Ett annat sätt att göra detta på är "if (!empty($_POST))". Men i ditt fall funkar detta galant
if (isset($_POST['first'])) {

    $dbServername = "";
    $dbUsername = "";
    $dbPassword = "";
    $dbName = "";

    $conn = mysqli_connect($dbServername, $dbUsername, $dbPassword, $dbName);

    if(!$conn){
        echo 'Not Connected To Server';
    }

    $first = mysqli_real_escape_string($conn, $_POST['first']);

    $query = mysqli_query($conn, "INSERT INTO savecolor (user_color) VALUES ('$first')");

    if ($query) {
        echo 'Inserted';
    } else {
        echo 'Not Inserted';
    }

}
?>