You can give users the ability to edit certain categories only by assigning them to new user groups, and then allowing each group access to manage the articles. Access to the administrator area will not be needed if you create a front-end edit menu.

Here's how to do it (works on Joomla 2.5 and 3, not sure about 1.5):

**1. Create new user groups**  
Log in to your admin area, and go to **Users** -> **Groups** -> **Add new group**. Give the group a name and set `Group Parent` to **Registered** (because registered users don't have any editing permissions to start with. Later we'll give them authorization to edit certain category).

![User groups][1]

Repeat this step for each category you want to give exclusive access to.

**2. Assign user groups to categories**  
Open the category manager (**Content** -> **Category Manager**) and open the category you want to grant access to.

Select the "Permissions" tab, and click the name of the user group that will be allowed to edit this category.

![Permissions tab][2]

Now change the dropdown lists under "Select New Settings" to allowed for all fields (as required) and click save. The red `Not Allowed` text should change to a green `Allowed` message.

Repeat as needed for other categories and corresponding user groups.

**3. Assign users to the groups**  
Finally, assign each user to the groups you created in step #1, depending on what category they are allowed to edit. This is done in **Users** -> **User Manager**.

Now your users will have access to edit only content from the categories you assign them to.

##Update##
Only user level `Publisher` and above are allowed to publish articles, thus this method does not allow users to publish their own articles (has to be approved by a higher-level user). It might be possible to overcome by setting the default value of the `state` column in the `com_content` database table to `1`, but I think I found a better solution:

**[User Article Manager][3]** is a component that gives you full control over users and articles. It's still in beta, and I just had a quick look, but you should be able to do exactly what you want, and allow users to publish only their own articles in specific categories.


  [1]: https://i.sstatic.net/C8iuY.jpg
  [2]: https://i.sstatic.net/sUbcS.jpg
  [3]: http://extensions.joomla.org/extensions/authoring-a-content/content-submission/18412