If I install an ATI or Nvidia video card, with 512 mb on it... do I need special partitions on my HDD?

Which video card is best for compiz fusion and other graphics?

When the ordinary RAM is free, how does that help the system?

Which is a good affordable brand and model of video card?

No, RAM on your graphics card has got nothing to do with your HDD

I'd prefer an nvidia card, just because they got better drivers than AMD/ATI. I think any card you can buy at a store should be good enough for compiz fusion. The question is, do you run any other 3d applications on your system? If so keep in mind, that by running compiz fusion it leaves you slightly less power for other 3d applications.

?? Sorry, I don't understand your question.

As mentioned before, I'd prefer nvidia. The model depends on what programs you want to run.

About RAM, it was just explained to me that video cards have their own memory, freeing the computers RAM. With a look at the back of a computer, and the monitor was connected to the Video card, with a special attachment, and the upper simpler plug, with the screw ins, that with no extra part, that the monitor cable could directly plug into, was left unplugged.

And a while back, I was advised that for compiz, you need at least 512 MB RAM and should have a partition of twice your RAM.

Yes and No: If you have an onboard Graphics Chip, it usually works with shared memory. That means, if your system has 512MB RAM, you can assign e.g. 64 MB of that to the graphics chip. Ergo there are 448MB RAM left for all other things (Operating System, Browser, Office, etc.) Therefore you could say a graphics card with it's own memory is 'freeing the computers RAM' though this term is actually not correct. But if your system is missing enough RAM a graphics card with 512MB definately wouldn't help, because it's not accessible by other computing tasks. It's exclusively for graphics operations.

If you're working on something that needs lots RAM, your system probably comes to the point, where it needs more RAM than there is actually available. In that case the system is freeing RAM by writing data temporarily to the harddrive. This is what Windows calls 'Virtual Memory'. Linux uses for that purpose a so called 'SWAP Partition'. It's true that a couple of years ago people advised you -as a rule of thumb- to create a SWAP Partition twice as big as your installed main memory. But nowadays... I have installed 2 GB RAM on my System and for me there is simply no need for a SWAP Partition anymore. Plus RAM is cheap, so nobody has to suffer from not enough RAM.

To make a long story short: If you're just for the Compiz and not playing Doom 3 or install Photoshop, just buy a cheap graphics card with 64MB and everything will be fine.

Answer to all of your questions;
Choose between Nvidia,ATI or Intel grafic cards because they are best supported and they are all good for whatever you will be doing.Sometimes one is better then another,but that depends for what and on what distro you are using it.
Don't wory about separate partitions for grafic cards,ordinary RAM and similar stuff,because that is close to no sense at all,you will learn about it if you start to learn about it.
I will just say that if a grafic card has more built in RAM,it's better for you.