To establish a connection between MySQL and PHP, you can use the MySQLi (MySQL Improved) extension or the PDO (PHP Data Objects) extension. 

Both options provide a set of functions and methods to interact with MySQL databases. 

Using MySQLi:

<?php
// Database credentials
$servername = "your_mysql_server";
$username = "your_mysql_username";
$password = "your_mysql_password";
$database = "your_mysql_database";

// Create a connection
$conn = new mysqli($servername, $username, $password, $database);

// Check the connection
if ($conn->connect_error) {
    die("Connection failed: " . $conn->connect_error);
}

echo "Connected successfully";

// Perform database operations here

// Close the connection
$conn->close();
?>

Using PDO:

<?php
// Database credentials
$servername = "your_mysql_server";
$username = "your_mysql_username";
$password = "your_mysql_password";
$database = "your_mysql_database";

// Create a connection
try {
    $conn = new PDO("mysql:host=$servername;dbname=$database", $username, $password);
    // Set the PDO error mode to exception
    $conn->setAttribute(PDO::ATTR_ERRMODE, PDO::ERRMODE_EXCEPTION);

    echo "Connected successfully";

    // Perform database operations here
} catch (PDOException $e) {
    echo "Connection failed: " . $e->getMessage();
}

// Close the connection (PDO automatically closes the connection when the script ends)
?>

Replace "your_mysql_server", "your_mysql_username", "your_mysql_password", and "your_mysql_database" with your actual MySQL server details.