Depends on the card. Most(not all) computers come with a basic graphics chip built in on the motherboard (Integrated Graphics). Those use some of your computer's own resources to operate. Like you may buy a computer that advertises 4 gb of ram but when you turn it on and check it says it only has 3.8 GB or something, the integrated graphics system has reserved some of it for it's use.
But there are also cards that go into expansion ports that are essentially an additional specialized motherboard just for handling graphics. Graphics Cards. All graphics systems have their own CPU (called a GPU) that process graphical information more efficiently and quicker than a CPU can. Graphics cards typically have a more powerful GPU, dedicated cooling, and dedicated ram built into the card itself, resulting in higher frame rates for graphics intensive games and programs than otherwise.
Generally the better you have the faster it can process those images. But different graphics systems can process images in different ways, sometimes causing images to appear slightly different on separate cards usually not in noticeable ways but some people will swear by specific types of cards. Newer and better cards can often make an image look better with better shader handling, better anti aliasing and such, but the graphics system can only work with what the program you are using can give it.
A cutting edge graphics card won't make DF look any better, but you'll be able to run more graphical intensive games at higher settings before you notice shuddering and poor framerates.
EDIT: And yea without any kind of graphics system you won't see anything but a black screen with some text on it at startup. Which is why most computers come with integrated graphics. With really only PCs marketed towards power users not always having them built in. Even if you have one built in, a graphics card can still be added to improve it. But this is where the depends on the card comes in. I have friends that have purchased graphics cards that are actually worse than the one integrated into their computer.