Order a MySQL table by two columns

Default sorting is ascending, you need to add the keyword DESC to both your orders:

ORDER BY article_rating DESC, article_time DESC

ORDER BY article_rating, article_time DESC

will sort by article_time only if there are two articles with the same rating. From all I can see in your example, this is exactly what happens.

↓ primary sort                         secondary sort ↓
1.  50 | This article rocks          | Feb 4, 2009    3.
2.  35 | This article is pretty good | Feb 1, 2009    2.
3.  5  | This Article isn't so hot   | Jan 25, 2009   1.

but consider:

↓ primary sort                         secondary sort ↓
1.  50 | This article rocks          | Feb 2, 2009    3.
1.  50 | This article rocks, too     | Feb 4, 2009    4.
2.  35 | This article is pretty good | Feb 1, 2009    2.
3.  5  | This Article isn't so hot   | Jan 25, 2009   1.

ORDER BY article_rating ASC , article_time DESC

DESC at the end will sort by both columns descending. You have to specify ASC if you want it otherwise