What does it take to play a game smoothly on a PC?

I currently have a 4 year old computer that has done great but it definitely isn't a gaming computer. I never had much of a problem with this because I only write and watch YouTube videos on this computer. Now I have gotten into gaming and I have recently grown interest in Minecraft and I would like to play the new(est) SImcity. I have played Minecraft and I play on the lowest settings and I still lag a lot and find it impossible to play. What exactly do you need to allow a game to run smoothly? Simcity is the most graphic heavy game I will be playing (these two games will possibly be the only games I will play on it), I won't be playing any sort of games like COD because I play those games my Playstation.

I would like to know the absolute minimum hardware I will need so when I go to buy a new system I know what I need. My uncle also works on computers for a living so if I would need to add anything to a computer he could do that for me.

Added (1). I actually have 3GB of RAM, I'm pretty certain it is the video card that is causing the freezing. I was wondering if buying a whole new system would be better but if a $50 - $100 fix would suffice then that would be great.

Added (2). @MArk I actually have a huge computer case that is around the same size as this comouter: http://www.tigerdirect.com/applications/SearchTools/item-details.asp?EdpNo=7246656&CatId=1510
It is just filled with junk parts that still work. I could always build a computer with that. I also just bought a mechanical keyboard and I already have a copy of Windows 7 and Windows 8. Is everything on this list really needed? Could some of these parts be down graded?

Since your PC is quite old in computer years, probably the best option would be a low-profile video card. The reason is that your CPU would most likely bottleneck a higher end video card, plus I'm sure the power supply can't hold the power requirements of new video cards.

Also consider expanding the RAM to 2Gb at the very minimum, with 4Gb as the optimal choice.

Link to a low-profile video card -> http://www.amazon.com/...004X8EO6Q/

This http://pcpartpicker.com/p/QwFRzy I know its expensive just save up