How to place two forms on the same page?

Well you can have each form go to to a different page. (which is preferable)

Or have a different value for the a certain input and base posts on that:

switch($_POST['submit']) {
    case 'login': 
    //...
    break;
    case 'register':
    //...
    break;
}

You can use this easiest method.

<form action="validator.php" method="post" id="form1">
    <input type="text" name="user">
    <input type="password" name="password">
    <input type="submit" value="submit" form="form1">
</form>

<br />

<form action="validator.php" method="post" id="form2">
    <input type="text" name="user">
    <input type="password" name="password">
    <input type="submit" value="submit" form="form2">
</form>

Give the submit buttons for both forms different names and use PHP to check which button has submitted data.

Form one button - btn1 Form two button -btn2

PHP Code:

if($_POST['btn1']){
    //Login
}elseif($_POST['btn2']){
    //Register
}

You could make two forms with 2 different actions

<form action="login.php" method="post">
    <input type="text" name="user">
    <input type="password" name="password">
    <input type="submit" value="Login">
</form>

<br />

<form action="register.php" method="post">
    <input type="text" name="user">
    <input type="password" name="password">
    <input type="submit" value="Register">
</form>

Or do this

<form action="doStuff.php" method="post">
    <input type="text" name="user">
    <input type="password" name="password">
    <input type="hidden" name="action" value="login">
    <input type="submit" value="Login">
</form>

<br />

<form action="doStuff.php" method="post">
    <input type="text" name="user">
    <input type="password" name="password">
    <input type="hidden" name="action" value="register">
    <input type="submit" value="Register">
</form>

Then you PHP file would work as a switch($_POST['action']) ... furthermore, they can't click on both links at the same time or make a simultaneous request, each submit is a separate request.

Your PHP would then go on with the switch logic or have different php files doing a login procedure then a registration procedure