How do I omit similar content in a for loop based on the first letter in title?
If you need only unique letters, you can build a new array and only merge first letters that are not already in the array like this:

{% set letterArray = [] %}

{% for alpha in craft.entries.section('advertisers').order('title asc') %}
  {% if alpha.title|first not in letterArray %}
    {% set letterArray = letterArray|merge([alpha.title|first]) %}
  {% endif %}
{% endfor %}

{% for letter in letterArray %}
  &lt;a href="{{ siteUrl }}{{ letter }}"&gt;{{ letter }}&lt;/a&gt;
{% endfor %}