Tuulikki theme uses the MailChimp plugin to create the form Subscribe to my newsletter.

To get the same email sign up style, just replace with this piece of code in the edit form

<div class="subscribe-box">

 <div class="subscribe-text">
 <h4>Get on list!</h4>
 <p>Sign up and receive the latest tips via email.</p>
 </div>
 <div class="subscribe-form">
 <input type="text" name="FNAME" placeholder="Name">
 <input type="email" name="EMAIL" placeholder="Email" required />
 <input type="submit" value="Subscribe" />
 </div>
</div>

To set the mailchimp wiget below slider go to: appearance > widget (see screenshot)