Tags are like micro-categories or index keywords for your posts. Your site should probably only have 7-12 categories, but you can have hundreds of tags. Clicking a tag name below your post will show all posts with the same tag, helping readers find similar content (most readers arrive at your site through searches for a specific topic). You add tags to your posts by typing the tag names in the Tags section below the post. You can also manage your tags by going to Manage > Tags.
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If tags are enabled in your theme, you will usually see them below the published post. If not, you can add a php tag to make it appear. (Because tagging is new with WordPress 2.3, many themes don't have the Tag template tag added by default.) If you go to Design > Theme Editor and click the single.php file, you'll see the code that generates your posts. Add to this file where you want the tags to appear (usually after the tag, which generates your post content).