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Definition
A membrane is a thin polymer layer in footwear or clothing that blocks liquid water but lets water vapour through. In shoes it sits between the upper and the lining, often as a bootie that fully wraps the foot. This keeps the shoe waterproof while it still breathes.
The physics
A raindrop consists of a vast number of molecules and is thousands of times larger than the membrane's micropores, so it cannot pass. Vapour from the foot consists of individual molecules far smaller than the pores. Transport is driven by the difference in temperature and humidity inside and out: vapour moves to where there is less of it.
History
Expanded polytetrafluoroethylene (ePTFE) was developed in the late 1960s, and in the 1970s it became the basis of Gore-Tex, the first commercial breathable waterproof membrane. Hydrophilic non-porous membranes and outdoor brands' own polyurethane membranes followed.
Nano-Tex
Nano-Tex is a nano-finishing technology for textile fibres, not a separate membrane. Nanoparticles form a fine fuzz on each fibre that repels water and dirt like a lotus leaf without closing the gaps between fibres. Such a finish ventilates well and protects from splashes, while a membrane is more reliable against downpours and puddles.
DWR finish
DWR (durable water repellent) is a coating on the outer material that makes water bead up and roll off. When it wears off, the upper wets out and forms a film of water that blocks vapour from escaping, so DWR is renewed periodically with special sprays.
Limitations
A membrane does not make a shoe fully waterproof: water can enter over the top of the shaft, and with very heavy sweating or high air humidity vapour cannot escape fast enough. Performance also drops when the upper is dirty or wet.