Il faut simplement enlever les passages à la ligne (br):
<style>
label, input{
display:inline;
}
</style>
<form action="" method="post">
<div class="form-group">
<label for="nom"></label>
<input id="nom" type="text" value="" name="nom"></input>
</div>
<br></br>
<div class="error-message"></div>
<div class="form-group">
<label for="email"></label>
<input id="email" type="email" value="" name="email"></input>
</div>
<br></br>
<div class="error-message"></div>
<div class="form-group"></div>
<br></br>
<div class="error-message"></div>
<p></p>
</form>