Is it possible to output month names in my native language? I'm doing this:

{# Get events from `events` section in desired sort order #}
{% set allEvents = craft.entries.section('events').order('eventsDate asc') %}

{# Loop through `allEvents` and print the list #}
{% for event in allEvents %}
    &lt;li&gt;{{ event.title }} - {{ event.eventsDate|date('j F, Y') }}&lt;/li&gt;
{% endfor %}


I guess I can do something with this: http://php.net/manual/en/datetime.formats.date.php, but I need to display months in my native language, which is slightly different from how Craft outputs it. 

For example I'd like it to say "7 des", not "07 dec", "15 mai", not "15th may" and so on.
Yes, the date() filter you use (see Twig docs) allows you to format dates with the same formating characters as the php date function takes. All language dependent parts are returned in the language of the current locale. I don't know if you can change that primary locale in the control panel, but you could do that in the DB.

As you noted in the comments to your question, you want the CP to be in english. This is possible even if you set the site's locale to another language; the site's locale and the user language for the CP are different things.