In order to run Flight Simulator X (FSX) to a reasonable level of graphical detail, you are going to need a pretty good PC.
The first thing to note is that FSX is processor intensive meaning that if you don't have a reasonable fast processor, it simply will not run smoothly no matter how great your graphics card is.
I, myself use an intel i7 920 quad core processor. The i7's are currently the best processors available for FSX. Why? Well mainly because Intel have designed them to be highly overclockable meaning that overclocking to speeds such as 4ghz is a piece of cake so long as you have a good cooler (these don't cost much at all).
You should also make sure that you are running enough RAM. I would recommend at least 6GB of quality ram such as that produced by corsair.
Next up, is the type of graphics card (GPU) you should invest in. I would recommend the Nvidia 460,480 or the new 580 model. I use an ATi 5870 which is pretty much the same speed as the Nvidia 480 however, it requires a lot more tuning to get good image quality. Therefore, I would recommend Nvidia over it.
To sum up, any of the i7's (depending on your budget) are the way to go if you want to match the visual experience you see in the videos shown on this website. I would also recommend you buy a custom built PC or build one yourself as opposed to buying from a manufacturer as they often lock their motherboards making it impossible to overclock the processor.
Below is a video to give you some tips on overclocking.
