form.html

<form name="sign_up" action="#" method="post">
  <label for="email">Email</label>
  <input type="text" name="email" />
  <label for="password">Password</label>
  <input type="password" name="password" />

  <button type="submit">Sign Up</button>

  <ul id="errors"></ul>
</form>

javascript.js

$("form").submit(function(e){
    e.preventDefault();
    $("#errors").value = "";
    var $e1 = "<li>Must be a valid email address</li>";
    var $e2 = "<li>Password must have at least one numeric character (0-9)</li>";
    var $e3 = "<li>Password must have at least one capital letter</li>";
    var $e4 = "<li>Password must be atleast 8 characters long</li>";
    var $email_valid = /.+\@.+\..+/;
    var $pass_num = /\d/;
    var $pass_cap = /[A-Z]/;
    var $pass_eight = /\w\w\w\w\w\w\w\w/;
    if($email_valid.exec(document.sign_up.email.value) === null) {
        $($e1).appendTo("#errors");
    };
    if($pass_num.exec(document.sign_up.password.value) === null) {
        $($e2).appendTo("#errors");
    };
    if($pass_cap.exec(document.sign_up.password.value) === null) {
        $($e3).appendTo("#errors");
    };
    if($pass_eight.exec(document.sign_up.password.value) === null) {
        $($e4).appendTo("#errors");
    };
});

style.css

ul#errors {
  color: red;
}