A dedicated video card (the type you could buy separately and install/replace) only has dedicated video memory. In fact all the memory it has on it is video memory, simply because it's a video card. There is no other kind on board. Naturally it is all used by the card alone, thus making it dedicated.

Now, when you have a video card integrated into the motherboard, things start getting tricky. The video card may have it's own memory, which no other component can access, that being the dedicated video memory. But it may also have access to overall system memory (the one usually called RAM), which is also used by your OS and programs and everything else.

That memory can also be divided in creative ways. A a chunk of it may be reserved for the system and the video card is unable to touch it. Another chunk of it may be reserved for the video card alone. The remainder could be used for whatever needs it the most at the moment.

The problems are as follows:

a) General memory is much slower than the one used on video cards, so even if you have a ton of it, you will still be doing things slower.
b) If you have a computer with an integrated video card, chances are it's a budget or otherwise a non-gaming computer, so you probably won't have a lot of memory anyway, forcing components to fight over it, again resulting in slow performance.
b) Depending on what software is running at the moment, the availability of memory for tasks such as video can be drastically different.

Actually, data can be transferred between the video card's memory and the main memory that the CPU can access directly, allowing a program that doesn't otherwise use the GPU to use its memory. Of course, I don't know of any example of a program that actually does this, but it's certainly possible. (I have read about using video memory as swap space on Linux, but that is a hack that doesn't get used much.)

In fact, part of video memory is actually mapped into the CPU's address space. In other words, a program running in kernel mode can actually treat part of video memory as if it were system memory (though it will be slower); I don't see any reason that one couldn't even store machine code there and execute it from video RAM! (Interestingly enough, I believe that, on the Commodore 64, many copy protection cracks did just that, though system architecture was very different back then; we didn't have GPUs capable of doing arbitrary calculations, for example.)