wondering how i can create a tagging system in php and with a mysql database, my initial thoughts were to create a row in the table where the articles are stored, named tags
You're describing a many-to-many relationship between Articles and Tags. You'd want to use an intermediate junction table to resolve that relationship.
Now, to find all articles that match a selected tag:
SELECT a.article_title
FROM Tag t
INNER JOIN Article_Tag_Xref atx
ON t.tag_id = atx.tag_id
INNER JOIN Article a
ON atx.article_id = a.article_id
WHERE t.tag_name = 'SelectedTagName'