The male member, seen at his most magnificent and in his full-sized glory, usually after being aroused by the sight of a beautiful female, and just before satisfying her longings. Particularly appropriate when used in reference to a cock of regal grandeur and majestic dimensions, standing stiffly erect and at attention so as to proudly display every inch of his full height.
Note the use of the pronoun "he" rather than "it." "Big George" is not simply a slang word for your dick, like penis or love torpedo, but a name given to him in his own right, as a personification of your sexual potency and energy. Therefore, a girl may talk about him to you, and may even talk to him instead of you, as in the examples given below.
As for the origin of the name, "Big" is rather obvious. The best guess as to "George" is that the use of the commonest name of Kings of England in modern times came to be given to the part of the body that epitomizes kingliness.
"Wow! Big George looks like he's really glad to see me."
"Oh, Big George, don't stop! You fill me up! You are the biggest cock I have ever had in me in my life! I want you to fuck me! I need you to fuck me! Oh, please don't stop fucking me! Fuck me! Fuck me! Fuck me! Fuck me! OHHHHHHHH!!!!!!!!!!!!!!!"