When I learning forcing theory, I am surprised by various foring definition,I want to know the original purpose, why define partial order in such way and some background material which can help me gain more intuition.

1. Cohen forcing.It was made by Cohen when he solve CON(ZFC+nonCH),It add many new reals,it is natural to think this if we consider the patial order form,But I heard Cohen used the boolean value model, if condider the boolean value model,how can I understand the intuition?

2. Random forcing.I just know it made by solovy.But I don't know the background.

3. Laver forcing.I just know laver forcing used to prove CON(ZFC+BC) by laver.

How about Sacks forcing,Hechler forcing, Mathias forcing,Miller forcing?