What is the purpose of multiple graphics cards?

I have issues fully grasping the purpose of putting two or more graphics cards in one system.

SLI and Crossfire claim that they combine the power of multiple graphic cards. If I understand the information on the NVIDIA site correctly, SLI combines the processing power but the memory does not increase at all. I assume it's the same for Crossfire.

I see, in theory, where this makes sense if you use two GPU's to drive one graphics output port, although it does seem a bit of a waste considering that you kind of don't even use the memory on one of the cards. But what happens if you connect one monitor to one card each?

What I can't figure out at all, however, is the purpose of just two or more non-linked graphics cards in a system. Say I have just one monitor, is the second card used for anything at all? If yes, used by what, and how can I check or influence it? But even if I had two monitors, connected to each card, how would that work? I seem to understand that not each monitor would be driven by each card entirely separate. If this was the case, there would be no room for linked cards at all, right? So there must be some limitation, but what is it?

As you can see, I'm slightly confused about it all. If one could explain the difference between linked GPUs and non-linked GPUs, the benefits and downsides, in single and multi monitor setups, I'd appreciate it a lot.

By the way, in my setup I have two Nvidia NVS-300 cards but just one monitor. So I started wondering if the second card is doing anything at all and if yes, what that is.

You can have multiple cards driving a single display (or a set of displays attached to one card) which is SLI/Crossfire or you can have multiple cards driving several completely independent displays which is non-SLI/Crossfire.

When you are using SLI/crossfire you are using the memory on both cards, but the mode of operation requires the same texture/object data to be duplicated on both cards.

What I can't figure out at all, however, is the purpose of just two or more non-linked graphics cards in a system. Say I have just one monitor, is the second card used for anything at all? If yes, used by what, and how can I check or influence it?

In this case you are right that they cannot be used to drive the same display, but they can be used completely independently. For example you could have one low power card driving the display and make use of a non-linked but higher powered card for things like scientific processing or GPGPU tasks.

But even if I had two monitors, connected to each card, how would that work? I seem to understand that not each monitor would be driven by each card entirely separate. If this was the case, there would be no room for linked cards at all, right? So there must be some limitation, but what is it?

In this case again you can use them completely independently again, making use of things like the video decoders to play multiple videos across many displays or using the 3D rendering power of each card to do separate 3D tasks. A lot of programs (games) can be told which monitor, and therefore which graphics card, you want them to run on. Sure you loose the combined benefit of both cards but if you were, for example, wanting to have two game demos running on two separate displays from the same machine then it is, as far as I know, perfectly feasible.

In addition to @Mokubais answer, graphics cards can do more than just display graphics — they are very good at simple, massively paralleled jobs (like password cracking, bitcoin mining, simulation calculations etc). In fact, NVIDIA has a whole API/language for passing information to the card to process and then getting the results back.