What does floating point biased notation have to do with 2's complement? During lecture the professor talked about Integer Compare for floats and kept mentioning two's complement. 

What's the relation between them?

All I understand is that two's complement can take us from a number to its negative.
IEEE 754 Floating point is a standard way of representing fractions in binary which does not use two's complement.  There's a discussion here about why that is.