wmorri is spot on for the form. When you set the form's method to post, the inputs are posted back to the server. The code in your email.php file would look like this:
$from = "email@example.com";
$to = "recipientEmail@domain.com";
$subject = "Thanks for your submission!";
$body = "Thank you " . $_POST['name'] . " for contacting us!";
$adminTo = "firstname.lastname@example.org";
$adminBody = "A user has requested information:<br />\n";
$adminBody .= "Name: " . $_POST['name'] . "<br />\n";
$adminBody .= "Message: " . $_POST['message'] . " <br />\n";
$headers = "From: $from\r\n";
$headers .= "Content-type: text/html\r\n";
mail($to,$from,$body,$headers); // sends the e-mail
mail($adminTo,$from,$adminBody,$headers); // send an administrative e-mail.
I didn't run that, but if memory serves, thats all you need aside from configuring the php.ini file to point to a default mail server to handle the request.
It sends two e-mails. One to the user letting them know you received their submission, and one to you informing you that you received one.
the headers (content-type) allows you to put html in the e-mail instead of plain text. the built-in "mail" function I used returns 1 on success and false on failure. You could put this in an if statement to check and see if the operation completed.
Hope this helps.
Also notice the $_POST variable. This is a global variable that exists on every post back to the server. It is an array whose contents are the form values that were posted back. The 'text' part that goes in the square braces corresponds to the name="" attribute of your input element in your form.
So $_POST['name'] in your case, pulls whatever value they entered into the <input type="text"> field where it's name="" attribute equals "name".
In wmorri's code:
<input type="text" name="name" />