High Melting Point grease is used for lubrication of areas that are likely to attain high running tempertures and can be used in bearings, etc - especially wheel bearings which can be subjected to heat from nearby brakes, etc. It is also suitable as a general purpose grease.
Lithium based grease is ideal for areas where there is risk of moisture or water penetration and is used extensively in the marine environment (especially boat trailer bearings) - it is ideal for bonnet/door locks and window regulators.
Silicon is usually available in liquid from and has relatively poor lubrication qualities but can be used to lubricate low speed movement areas rubber or plastic - ideal for window tracks and squeaky rubber or plastic bushes.
Graphite in powder form similarly has poor high speed lubrication qualities on its own and is used in areas where a grease or liquid wouldn't be suitable and is likely to attract dirt or dust- it is used extensively in the lubrication of locks and is impervous to low temperatures - generally it has very poor anti-corrosive qualities. Most decent locksmiths will sell small flexi puffers for blowing it into keyholes, etc. The downside is that it can attract moisture due to its powder format.
Molybdenum Disulfide Grease - "black grease" as it is commonly known contains a type of graphite and is used mainly in drive couplings and in areas of relatively slow moving components and is also good at resisting moisture and has excellent anti-fling properties - in other words it sticks like **** to the perverbial blanket
Copper grease isn't actually a grease at all and is primarliy an anti-seize and anti-corossive compound. It can normally stand high temepratures but is a poor high speed lubricant and if used in high speed bearings can cause early failure. It is primarliy designed as a corrosive inhibitor. Whilst many mechanics seem to use it on the back of brake pads it is not generally recommended for such use by brake manufacturers as it can attack brake piston seals and gaitors. A special grease is available for such use and does not contain any metals or acids and even at that it should be used sparingly. Mintex Ceratec is an example of so-called brake grease. Also anti-squeal shims are available to reduce brake squeal and vibration.
Many modern greases and lubricants can be a mixture of any of the above but as yet no-one seems to have make a one type suits all product yet.
Horses for courses as they say
I am no expert on grease or lubricants and I have written the above based on my many years of experience both in the vehicle and marine engineering. No doubt should anyone wish to add any helpful comments please feel free to do so