Adding and managing categories and tags is done from the posts sub menu.
Hover over posts and click categories.
You will see a list of categories that have been created on the site along with a count of the number of posts that are within that category.
To add a new category fill in the new category form.
The slug field is a website URL friendly version of whatever you type in the category name box. When you click the blue add new category button wordpress will automatically fill this form, although you can provide one yourself. The basic rule of a slug is to replace any dashes or punctuation with a dash to ensure the URL works well.
If you want the category to be a child of a previously created category choose the parent from the dropdown.
Give the category a description if you want to and will display it on the page.
Click the add new category button and your new category will appear to the right hand side.
You may carry out a bulk delete action on categories or delete them singly. This won't delete any posts in these categories.
Moving to the tags section by hovering over posts and selecting tags you will see this operates in exactly the same way as categories apart from the fact that tags are not hierachical.
Tags are mostly used to give extra context to blog posts and should be used sparingly. Each tag you create, as with categories, will create an associated page in wordpress listing posts. Some search engines will view tag pages as low quality and ignore them.