Refer to figure.
An aircraft's glide angle for longest range is decided by it's lift to drag ratio, and given that lift just needs to equal weight (slightly less than weight in a glide - but still a constant value), the only deciding factor is drag. Therefore, an aircraft glides the longest distance (the shallowest glide angle) when drag is lowest, so at V_MD, minimum drag speed.

Should, for any reason, the aircraft want to remain in the air for a longer time, they can descend at the best glide endurance speed. By flying slightly slower than V_MD, the aircraft has slightly more drag, so a steeper descent angle, but is flying down that angle at a much reduced speed, overall leading to a lesser rate of descent, and a longer time in the air.

This reduced speed for best endurance in a glide is V_MP, the speed for minimum power required. It is not affected by what type of engine the aircraft has, because this is a glide descent, so the engine is not functioning. It is also not affected by headwinds or tailwinds, as they do not increase or decrease rate of descent, they only change where the aircraft will eventually meet the surface, not how long the descent will take.