Your graphics card indeed looks like it's pretty old and hasn't got the power to handle today's games, be it bit older games too, especially in HD/full HD.
It's definitely not your cpu, it's fine.
You could try updating graphic drivers
from ati, if they are not up to date (10.6 version if im not mistaken).
If that doesn't work it's probably that your card lacks the punch.
Can you list the games that it happens in?