BesselJ and BesselY are the Bessel functions of the first and second kinds, respectively.  They satisfy Bessel's equation:

BesselI and BesselK are the modified Bessel functions of the first and second kinds, respectively.  They satisfy the modified Bessel equation:

HankelH1 and HankelH2 are the Hankel functions, also known as the Bessel functions of the third kind.  They also satisfy Bessel's equation, and are related to BesselJ and BesselY by