The other users already told you what doesn't quite fit. I would contribute with some suggestions and code too.
- Use the object oriented mysqli. I recommend to use PDO though.
- Use prepared statements to avoid SQL injection. Read this too.
- Add the proper "meta" tags, followed by the "title tag right after them.
- No content output should be found inside the html "head" part. It belongs to the "body" part.
- Forget the <center>tag and use only the "text-align: center" (+ "position: relative" if needed) css rule instead. Read Centering in CSS: A Complete Guide.
- Require "phpsearchcode.php" on top of the "search.php" code.
- In the "phpsearchcode.php" fetch all data into an array ($results) and close the result, the statement and, eventually, the connection right after. Then you will loop through the $results array in the search.php html code. By doing this you avoid mixing database fetching code with the html output code (as you did). Principle: Fetch data above, but display the results down under. So to say. This way you are also avoiding printing html tags from PHP (using "echo" or similar).
- Add a container ("results-container") to hold the results rows. So that you can position the container as a whole where you wish (by maybe applying "top: 55%" and "width: 20%" rules to it - instead of it's content rows). The content rows themself should become only "padding", "margin" and "text-align: center" rules. The "text-align: center" will apply to the text inside the "results" div.
Good luck.
phpsearchcode.php
<?php
// Db configs.
define('HOST', 'localhost');
define('PORT', 3306);
define('DATABASE', 'R');
define('USERNAME', 'root');
define('PASSWORD', '');
/*
 * Error reporting.
 * 
 * @link http://php.net/manual/en/function.error-reporting.php
 * @link http://php.net/manual/en/function.set-error-handler.php
 * @link http://php.net/manual/en/function.set-exception-handler.php
 * @link http://php.net/manual/en/function.register-shutdown-function.php
 */
error_reporting(E_ALL);
ini_set('display_errors', 1); // SET IT TO 0 ON A LIVE SERVER!
if (isset($_POST['search'])) {
    /*
     * Enable internal report functions. This enables the exception handling, 
     * e.g. mysqli will not throw PHP warnings anymore, but mysqli exceptions 
     * (mysqli_sql_exception). They are catched in the try-catch block.
     * 
     * MYSQLI_REPORT_ERROR: Report errors from mysqli function calls.
     * MYSQLI_REPORT_STRICT: Throw a mysqli_sql_exception for errors instead of warnings. 
     */
    $mysqliDriver = new mysqli_driver();
    $mysqliDriver->report_mode = (MYSQLI_REPORT_ERROR | MYSQLI_REPORT_STRICT);
    /*
     * Create a new db connection.
     * 
     * @see http://php.net/manual/en/mysqli.construct.php
     */
    $connection = new mysqli(HOST, USERNAME, PASSWORD, DATABASE, PORT);
    // Get submitted values.
    $search = '%' . $_POST['search'] . '%';
    /*
     * The SQL statement to be prepared. Notice the so-called markers, 
     * e.g. the "?" signs. They will be replaced later with the 
     * corresponding values when using mysqli_stmt::bind_param.
     * 
     * @link http://php.net/manual/en/mysqli.prepare.php
     */
    $sql = 'SELECT username 
            FROM userinfo 
            WHERE username LIKE ?';
    /*
     * Prepare the SQL statement for execution - ONLY ONCE.
     * 
     * @link http://php.net/manual/en/mysqli.prepare.php
     */
    $statement = $connection->prepare($sql);
    /*
     * Bind variables for the parameter markers (?) in the 
     * SQL statement that was passed to prepare(). The first 
     * argument of bind_param() is a string that contains one 
     * or more characters which specify the types for the 
     * corresponding bind variables.
     * 
     * @link http://php.net/manual/en/mysqli-stmt.bind-param.php
     */
    $statement->bind_param('s', $search);
    /*
     * Execute the prepared SQL statement.
     * When executed any parameter markers which exist will 
     * automatically be replaced with the appropriate data.
     * 
     * @link http://php.net/manual/en/mysqli-stmt.execute.php
     */
    $executed = $statement->execute();
    /*
     * Get the result set from the prepared statement.
     * 
     * NOTA BENE:
     * Available only with mysqlnd ("MySQL Native Driver")! If this 
     * is not installed, then uncomment "extension=php_mysqli_mysqlnd.dll" in 
     * PHP config file (php.ini) and restart web server (I assume Apache) and 
     * mysql service. Or use the following functions instead:
     * mysqli_stmt::store_result + mysqli_stmt::bind_result + mysqli_stmt::fetch.
     * 
     * @link http://php.net/manual/en/mysqli-stmt.get-result.php
     * @link https://stackoverflow.com/questions/8321096/call-to-undefined-method-mysqli-stmtget-result
     */
    $result = $statement->get_result();
    /*
     * Fetch data and save it into an array - to be looped through in the later html code.
     * 
     * 
     * @link http://php.net/manual/en/mysqli-result.fetch-all.php
     */
    $results = $result->fetch_all(MYSQLI_ASSOC);
    /*
     * Free the memory associated with the result. You should 
     * always free your result when it is not needed anymore.
     * 
     * @link http://php.net/manual/en/mysqli-result.free.php
     */
    $result->close();
    /*
     * Close the prepared statement. It also deallocates the statement handle.
     * If the statement has pending or unread results, it cancels them 
     * so that the next query can be executed.
     * 
     * @link http://php.net/manual/en/mysqli-stmt.close.php
     */
    $statement->close();
    /*
     * Close the previously opened database connection.
     * 
     * @link http://php.net/manual/en/mysqli.close.php
     */
    $connection->close();
}
search.php
<?php
require_once('phpsearchcode.php');
?>
<!DOCTYPE html>
<html>
    <head>
        <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1" />
        <meta name="viewport" content="width=device-width, initial-scale=1, user-scalable=yes" />
        <meta charset="UTF-8" />
        <!-- The above 3 meta tags must come first in the head -->
        <title>RS search</title>
        <style type="text/css">
            /**************/
            /* Base rules */
            /**************/
            html,body {
                background-color: #282C35;
                color: white;
                margin: 0px;
            }
            a {
                text-decoration: none;
                color: #C4C4C4;
            }
            input {
                background-color: #282C35;
            }
            /****************/
            /* Layout rules */
            /****************/
            .results-container {
                /* ... */
            }
            .results {
                background-color: #36393E;
                width: 20%;
                text-align: center;
                padding-bottom: 30px;
                padding-top: 10px;
            }
            #logo {
                top:40%;
                position: absolute;
                right: 33%;
            }
            #rat {
                background-color: white !important;
                border: 0px;
                transform: translate(-50%,-50%);
                position: absolute;
                top: 50%;
                right:30%;
                padding: 20px;
            }
            #bug {
                border: 0px;
                background-color: white;
                transform: translate(-50%,-50%);
                position: absolute;
                top: 50%;
                padding: 20px;
                width:350px;
            }
        </style>
    </head>
    <body>
        <img src="rs.png" id="logo"/><br />
        <form action="search.php" method="post">
            <input type="text" name="search" id="bug">
            <input type="submit" name="submit" value="search" id="rat">
        </form>
        <?php
        if (isset($_POST['search'])) {
            ?>
            <h3>
                Results list
            </h3>
            <div class="results-container">
                <?php
                if (isset($results) && $results) {
                    foreach ($results as $row) {
                        $username = $row['username'];
                        ?>
                        <div class="results">
                            <?php echo $username; ?>
                        </div>
                        <?php
                    }
                } else {
                    ?>
                    <div class="noresults">
                        No users found!
                    </div>
                    <?php
                }
                ?>
            </div>
            <?php
        }
        ?>
    </body>
</html>