Yup the intel HD integrated graphics is your problem.
Just for a reference point:
Your GPU: http://www.videocardbenchmark.net/gpu.php?gpu=Intel+HD+4000
A 6 generation old budget gaming card: http://www.videocardbenchmark.net/gpu.php?gpu=GeForce+GTX+260&id=17
What FFXIV recommends you use (gtx 660) http://www.videocardbenchmark.net/gpu.php?gpu=GeForce+GTX+660&id=2152
The intel HD is a really cheap solution for people who mainly play platformers and single player games on low settings. You might get away with some modern single player games running decent.
MMORPG games are usually extremely taxing on GPU's even if their graphics are not top notch (this game is both) because of all the other players on screen and their skills ect.
Get yourself a new GPU, which will most likely require you to get a new Power Supply Unit to power it (factory built computers come with the BARE minimum PSU)
Its always a good idea to know what GPU you have and where it stands performance wise...it will stop you from buying a game that wont play or will run like crap (granted you read system requirements) Edited, Sep 6th 2013 9:38pm by Strangerous