What is the connection between HDL and cholesterol?

A measure. Cholesterol (c) is actually transported through the blood as a constiuent of lipoproteins. These are associated with atherosclerosis; ldl-low density liporoteins-causing and hdl-high density lipoproteins-protecting against. When c l is fractionated and measured the LDL cholesterol correlates to the amount of LDL and the HDL cholesterol correlates to the hdl. Thus c provides an indirect measure.
HDL Removes All Fats. Ldl & HDL are lipoproteins, i.e. Protein particles which carry all fats in the water outside cells. Large-hdl removes fat from cells overloaded. Ldl is always in nmol/l, best <700, HDL in µmol/l, best large-hdl >10. See NMR particle test, <$100. Cholesterol is a fat made by every body cell, promoted because cheaper to measure but not best test, ldl-c is calculated, not measured & often misleading.