How do I call the validation when the textbox looses focus? Eg.

@using (Html.BeginForm()) {
    @Html.ValidationSummary(true)

    <div class="editor-label">
        @Html.LabelFor(model => model.Name)
    </div>
    <div class="editor-field">
        @Html.EditorFor(model => model.Name)
        @Html.ValidationMessageFor(model => model.Name)
    </div>
}

public class RegisterViewModel
{
    [Required]
    [StringLength(5)]
    public string Name { get; set; }
}

Also, How can I check if the name exist already by calling an action?

You can enable eager evaluation by using setDefaults() to override the onfocusout property with a function that always evaluates the blurred element by calling the valid() method.

<!-- language: lang-js -->
// enable eager evaluation
$.validator.setDefaults({
    onfocusout: function (element) {
        $(element).valid();
    }
})

Demo With Eager Evaluation:
<sup>(Will show required after tab out)</sup>

<!-- begin snippet: js hide: true --> <!-- language: lang-js -->
// enable eager evaluation
$.validator.setDefaults({
  onfocusout: function (element) {
    $(element).valid();
  }
});

$("form").validate();
<!-- language: lang-css -->
input.error {
  border-color: red
}
<!-- language: lang-html -->
<script src="//cdnjs.cloudflare.com/ajax/libs/jquery/2.1.3/jquery.js"></script>
<script src="//cdnjs.cloudflare.com/ajax/libs/jquery-validate/1.13.1/jquery.validate.js"></script>

<form >
  <input type="text" name="Name" required /><br/>
  <input type="text" name="Email" required /><br/>
  <input type="submit" value="Submit"/>
</form>
<!-- end snippet -->

Normal Demo
<sup>(Won't show required until submit)</sup>

<!-- begin snippet: js hide: true --> <!-- language: lang-js -->
$("form").validate();
<!-- language: lang-css -->
input.error {
  border-color: red
}
<!-- language: lang-html -->
<script src="//cdnjs.cloudflare.com/ajax/libs/jquery/2.1.3/jquery.js"></script>
<script src="//cdnjs.cloudflare.com/ajax/libs/jquery-validate/1.13.1/jquery.validate.js"></script>

<form >
  <input type="text" name="Name" required /><br/>
  <input type="text" name="Email" required /><br/>
  <input type="submit" value="Submit"/>
</form>
<!-- end snippet -->