Indeed, what's needed for a user to submit a new content item from the frontend, and how he/she should proceed?
First of all the user must have the privileges corresponding to the authorization level of "author" or above.
Then, of course, must be logged in . Obviously, the site admin must allow submission of new items.
Once logged in and selecting the corresponding link - let's say "Add News Item" from the User menu, an author can add his article to the allowed/corresponding section.
Alternatively, on each menu item of Content Table type the user will see a "New" tag below the table of available items. Clicking on it, will open the same editing window, with the corresponding section pre-selected. From here the author can write the new item, tweak it and save it.
Depending on the site settings, the new item will show up on the site immediately or after approval by another site administrator, with corresponding credentials.
From the above it's obvious, what needed to be done on admin side: Just create the link of appropriate type (Table Content Category) for the content sections/categories you want to allow your "author" level users to be able to submit new content items.
The default Add News Item also can be tweaked or new similar items can be created, for each categories the admin want to allow his users to submit articles from the frontend. The recipe it is the same for all the categories - in the "usermenu", create a new menu item: Submit - Content then select the desired section.
The only update here, since the Joomla 1.7 was launched, that instead of sections/categories, we are talking about categories alone!