I am going to put in order of most important to least important parts for a computer that performs well with high intensity projects (aka gaming).
1.(completely optional but most important for those that want the best machine) A new Tower Case is the most important part. So you ask how is this THE most important part, well if you want your computers case to be able to hold all your new parts this is the most likely a important and cheap piece. Also when we get to other parts you will most likely run into generic parts that have ALL the cables so it can work in any computer making it take two times more room then it should...
2. A new Power Supply is essential, if you don't want your computer burning out from not having enough power to run your new parts then you should get a power supply that is about 500+ watts. Better performance usually means greater power usage, as time goes on parts will become more power efficient so once you upgrade this it will probably be good to use all the way until it breaks.
3. A better Video Card the first part that actually directly deals with enhancing performance. The only reason why it is third is because without 2 this won't run right and 1 (though not required) the video card will be a completely new part and is not replacing a part so room is needed. The reason why it does not need to be bought to run a computer is because every computer has a Onboard Video Card that is efficient enough to support a default monitor that is attached to it and is rarely able to handle new games that are being put out.
4. A better Processor for the most simple reason is to be able to process more information. With the first three parts you will be able to see pretty and detailed pictures to be displayed on screen, but without a good processor you will not be able to see fluid movement on games or movies. Most computers have a good processor since these are generally a cheap part that drastically improves performance which is why its the forth most important part.
5. More RAM is another good thing to get. RAM is a part that assist the processor so it can more easily access information from the Hard Drive (the part that holds the information). RAM transfers information from the hard drive and attaches it to itself so it can access the information more quickly. When the processor has to get information from the hard drive it has to go through a set sequence to be able to access the information it wants. But RAM holds the information that it needed to access so it can more quickly get to it. I might do a post to better describe RAM but this is the just of what it is.
6. For now the last I can think of is a new Motherboard. The motherboard is the part that holds and connects all the parts of the computer. If you think you need a new one, the motherboard is what determines what type of part you need to be compatible. When installing new parts the motherboard is the part that tells you the capacity of parts that you can put into your computer. Even though this is last on the list the motherboard is the first part that you would want to buy if you are building a computer from scratch.
This will be it for now and I will probably do a post on how to install parts next. Keep on requesting so I can deliver.