Chemiosmosis is one of the processes a cell uses to extract energy from food. It is the final step of cellular respiration (hereafter referred to as glucose for simplicity) in many multicellular organisms, humans included. After glycolysis and the Krebs cycle break down glucose into smaller organic compounds with minimal net gain in energy, the electron transport chain establishes a proton gradient. The proton gradient's potential energy is converted into kinetic energy which facilitates ATP synthesis. This conversion is the process of chemiosmosis.
. . . What?
The best way to grok chemiosmosis is through analogy. Consider an altitudinous mountain. A river flows freely from the top of this mountain to its bottom, thanks to gravity's persistent help. Now build a dam somewhere along this river. This dam will have (say) a hundred little openings through which water passes. In each of these openings is a wheel. As water flows through an opening, the wheel spins. The wheel's spinning can help us do work, such as grind wheat into flour. Eventually, without some sort of outside help, the river will run out of water. So once in a while it rains on the top of the mountain, glaciers melt, aliens refill the supply, or whatever. Keep in mind the primary goal of the dam is to grind wheat into flour.
As it turns out, chemiosmosis is a very similar process. The mountain is analogous to a mitochondrion -- a microscopic organelle that every animal cell owns. The mitochondrion is the center for nearly the entire process of cellular respiration. The mitochondrion's purpose is to convert food (glucose) and ADP into energy-rich ATP and waste (H2O and CO2). The mitochondrion is split into two parts by a membrane: an outer compartment and an inner compartment. In our example, the outer compartment would be everything above the dam, and the inner compartment would be everything below the dam. The dam itself is the membrane, which is creatively named the "inner membrane." The outer membrane of the mitochondrion is what separates it from everything else in the cell; we can safely disregard it for didactic purposes.
The river's water ultimately comes from rain, glaciers, or whatever it was you decided. This is when the electron transport chain (ETC) has a part. The ETC produces the "river water" of chemiosmosis: it places H+ ions in the outer compartment of the mitochondrion (above the dam) and OH- ions in the inner compartment (below the dam). For the chemically illiterate, H+ and OH- combine to form H2O which is actually a waste product of this process. The inner membrane (the dam) is impermeable to both H+ and OH- (just like water cannot go through the concrete dam). As everybody who is anybody knows, opposites attract. So the H+ and OH- ions would really prefer there to be no dam in their way. They would even go so far as to spin wheels to meet up with one another. Inside each of the little openings in the mitochondrion is an F0F1 particle (the wheel). This F0F1 particle is an enzyme that catalyzes the synthesis of ATP. As the H+ ion goes through the F0F1 particle, energy is harvested which is used to produce ATP. The OH- ions do not pass through the F0F1 particle.
What would happen if not everything went according to plan? What if H+ and OH- ions could freely pass through the inner membrane? Think what would happen if water could permeate the dam's walls. Sure, some energy might be produced, but you would simply be wasting your time. What would happen if the "river water" were not replenished? Disruptions in the electron transport chain (which are what poisons usually do) would cause chemiosmosis to all but stop. If chemiosmosis stops, the organism would run out of ATP, which is necessary to nearly every bodily function, and die.
Peter Mitchell described the process of chemiosmosis in 1961 and subsequently won the 1978 Nobel Prize in Chemistry .
My AP Biology class
Campbell, Neil; Reece, Jane; and Mitchell, Lawrence. Biology. 5th ed. Menlo Park, CA: Addison Wesley Longman, Inc., 1999.
I developed this analogy independently but I have come to find out that it is of course not original. Some other sources (however incredible) use this analogy very briefly such as http://www.biology.lsu.edu/webfac/dlongstreth/biol12014f02/lecture_12.htm