I feel like you've already answered your questions.
- One sign up form = one form tags.
- Multiple forms = many form tags.
Just don't nest them.
EDIT
Form tags are meant to hold a variety of fields (i.e. input tags) that you will eventually pass to a target URL using a GET or POST request.
Take this login form, for example:
<form action="login.php">
<input id="name" type="text" name="name">
<input id="passwd" type="password" name="passwd">
<input type="submit" value="Login">
</form>
This has a login, a password, and a submit button. When the "Login" button (type = "submit") is pressed, the form will take that information and send it to another URL (in this case, "login.php", and that URL will handle the information accordingly (e.g. validate, sign you in, display captcha).
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