Depending on the processor you are using and the games you want to play this might answer may vary, anyways For Intel: If you are using core i3 or higher you can play easily any high resolution games with having a 2gb graphics NVidia Card. read more
for gaming you need to look for at least a 3GB or more and I'm telling this to you with personal experience of using a laptop with 2GB graphics which runs games at the lowest possible setting to just get to a playable state. read more
A normal desktop /laptop monitor would be good enough for a 1 GB graphic card. If you are running games on a 1080p screen, you probably need a better one. 2 GB and 4 GB graphic cards are mostly used by people with multiple-monitor setup, serious gamers and professionals plug in more than one high-res monitors, which , requires more amount of graphic memory. read more