My Intel (DG33) motherboard has 256MB built in graphics memory. I want to buy a monitor whose resolution is 1080*1920. To support this resolution, do I have to buy a graphics card or built in motherboard memory would be enough?
It depends on what you'll you be doing with your system. For gaming, this is definitely not enough. If you're only browsing and doing stuff like that, you should be fine.
However you can get a real cheap graphics card for about 50$ these days.