difference between single core CPU and quad core CPU?

Well you are almost right

and this is why they are faster than dual core

there are other CPU which are actually single core but work like a dual core :P

does increase the speed and better performance

Single core processors [One brain that utilizes one part of the brain] can only do one task at a time.

Quad core processors [One brain that thinks like 4 brains or utilizes all the parts of the one brain] can do multiple tasks at the same time without a sweat.

Yes, it's like saving spaces for cpu. the more core you have. the lesser lag you play on games.

More cores will give you more total power and better multitasking abilities.

Multitasking: http://en.wikipedia.org/wiki/Computer_multitasking