Here, we are going to use the WordPress’s wp_signon function to authenticate a user login.
See the following PHP codes to learn how to use it.

if($_POST){
    //We shall SQL escape all inputs
    $username = $wpdb->escape($_REQUEST['username']);
    $password = $wpdb->escape($_REQUEST['password']);
    $remember = $wpdb->escape($_REQUEST['rememberme']);
 
    if($remember) 
        $remember = "true";
    else 
        $remember = "false";

    $login_data = array();
    $login_data['user_login'] = $username;
    $login_data['user_password'] = $password;
    $login_data['remember'] = $remember;

    $user_verify = wp_signon($login_data, true); 
 
    if(is_wp_error($user_verify)){
        echo "<span class='error'>Invalid username or password. Please try again!</span>";
        exit();
    }else{
        echo "<script type='text/javascript'>window.location='". get_bloginfo('url') ."'</script>";
        exit();
    }
}

//Embed the login form here