It seems like your form and fields are missing labels and (more importantly) names. I modified your HTML, this should work (but I haven’t tested it):
<form name="contact" method="POST" data-netlify="true">
<div class="fields">
<div class="field name">
<label for="name">Name</label>
<input type="text" id="name" name="name" placeholder="Name" required>
</div>
<div class="field email">
<label for="email">Email</label>
<input name="email" id="email" type="email" placeholder="Email" required>
</div>
<div class="field">
<label for="subject">Subject</label>
<input id="subject" name="subject" type="text" placeholder="Subject" required>
</div>
<div class="field textarea">
<label for="message">Message</label>
<textarea id="message" name="message" cols="30" rows="10" placeholder="Message.." required></textarea>
</div>
<div class="field">
<div data-netlify-recaptcha="true"></div>
</div>
</div>
<div class="button">
<button type="submit">Send message</button>
</div>
</form>