Categories are used to organize posts by topics. They can also be used to group posts together on pages of your site. For instance, if you have categories for each of the grades at your school, you can setup a page that will display only that grade. A “1st Grade” page would display all the news in the 1st Grade category. This page could then have an introduction at the top, listing the teachers for that grade (for example). Your post should have one category, unless you require adding it to more than one so that it’s included in a slider, etc.
Tags are ways for visitors to search for content by keywords. Search will do this automatically, but by using tags, any post tagged with a certain keyword will come up higher on the search results. Also, a tag cloud can be added to your site that will display the tags, and show by size, the popularity of that tag. For an example of a tag cloud in action, see the bottom left of this page.