How to list the latest 3 entries per category per section?
You need to loop over the sections. In each section, you then loop over the categories, and display your entries inside them. Something like this.

{% set categories = craft.categories.group('groupHandle').find() %}
{% for section in craft.sections.getAllSections() %}
    {% if section.type == 'channel' %}
        &lt;h1&gt;{{ section.name }}&lt;/h1&gt;
        {% for category in categories %}
            &lt;h2&gt;{{ category.title }}&lt;/h2&gt;
            {% for entry in craft.entries.section(section).relatedTo(category).limit(3).find() %}
                &lt;a href="{{ entry.url }}"&gt;{{ entry.title }}&lt;/a&gt;
            {% endfor %}
        {% endfor %}
    {% endif %}
{% endfor %}


This is a pretty quick and dirty example. I pull the categories ahead of time so I can reduce the number of database calls. I suggest using the cache to speed things up.

The template docs (especially the ElementCriteriaModel) are handy reading materials.