Tags are a way to classify the site content in an unordered fashion. It's a powerful way to group entities of any type.
When viewing an entity which can be tagged (look here to see which types can be tagged), you should see the tags box appear displaying tags applied to this entity (if not, check your user preferences, and permitting to handle them.
To add a tag, click on the add tags link in the box to get the add tag field. As you type in this field, you should see appear existing tags matching characters you types if any. You can then:
Eventually you can add multiple tags by separating them with a comma.
Then type Enter or click the validate link to add the tag, or click the cancel link to cancel the operation.
To remove a tag, simply click on the [-] sign to the left of it.
Note that if you wish, you can change your preferencece to display them in a component in the top or in the bottom of the page instead of the box. Provided functionalities are the same, except for quick link to remove a tag.