TITLE: What is a $P$-name in forcing theory
QUESTION [16 upvotes]: I am having troubles understanding what is a $P$-name is forcing theory and what is the purpose of this term in the forcing tecnique.
Is there any simple way to explain this term.
If there was I would be greatfull if you shared it with me.
I read about it in Jech's book and in Kunnen's book but I am drowning in the technical detains.
I did get the terms "generic filter" or Martin's Axiom. But this one...:(
The only idea coming to my head is this:
M is a transitive countable model. $G \subseteq M$ is $P$-generic. $G \notin M$. We want to add $G$ as an element. But, $G$ may not be transitive. So, we add all the $(\alpha,p)$ where $\alpha \in M$, $p \in G$. 
Another question, Can I assome that all the elements of $M$ are sets of ordinals?
(M is a countable transitive model of ZFC)
Thank you!

REPLY [4 votes]: Here is an example of mine for a $\mathbb P$-name for $A = \{ 0,2 \}$. What do you think?