What are the different methods of knitting with two needles?
The German method is also called Continental Knitting. 
It is a method of knitting with 2 needles, and a type of style of how you hold your yarn and needles. You would hold the yarn in your left hand and your right hand will be on your working needle.

http://www.knittinghelp.com/video/play/the-knit-stitch-continental-continental

Another style is called English Knitting, also known as right-hand knitting or throwing. The yarn is carried in your right hand instead.

http://www.knittinghelp.com/video/play/the-knit-stitch-english-english

So, the main difference between the two is the way the yarn is wrapped around the right needle before it’s pulled through to knit a stitch. The choice between them though is merely preference for the most part as they both create the same resulting fabric. But here is more detailed post explaining the differences:

http://www.craftsy.com/blog/2014/02/english-vs-continental-knitting/

As for French knitting, as far as I know it is a type of tubular knitting, not 2 needles, which requires a knitting spool instead.