Custom content type add form I am brand new working with Drupal 7 and I am working on a professional project and I would like some help to achieve some requirements.

We have some content types like associations. Authenticated users may add new associations after administrators validation.
What is the best approach to display custom forms that are able to insert data as content types?
This forms must appear in a non administrative environment, so, I don't know if there is something to export Drupal add/content form and display it to the authenticated user.

I will appreciate any help.
Cheers!
Any administration page such as /node/add will be displayed in the selected theme, as long as the user has the permission 'View the administration theme' off (and of course has the relevant permission on, such as to create nodes for the specific type).

You wouldn't really need to create a custom form for that, the standard node create form may be sufficient. Of course you could, you would have to read the documentation such as this.

As far as the first requirement about validation from administrators, the modules Workflow and/or Rules would be useful.