Martensite is most often used to describe a type of compound of iron and carbon which contributes hardness and toughness to steel. Martensitic steels are strong, hard, tough and very useful for weapons, such as swords. However, among materials scientists, Martensite is used in a more general sense to describe crystals which have changed geometries following rapid cooling.
Martensite was discovered by a German microscopist, Adolf Martens around 1890. He was studying prepared slides of various steels, and observed banded crystal structures on some of the hardest, toughest steels. This characteristic pattern is now one of the identifiers of a martensitic crystal structure.
The steel used in swords and suchlike is martensitic, which means it has these martensite structures within it, and they give the steel its hardness and toughness. But every swordsmith knows the way to make a good sword is to heat the steel to a particular temperature, and then quickly quench it in water--or sometimes blood.
Just as an aside, metallurgists use a phase diagram to understand which compounds and crystal structures are stable at the various temperatures and concentrations of each alloying ingredient. The phase diagram for steel (iron and carbon) is very complex, with many different chemical combinations and crystal structures stable at varying places on the diagram.
What happens during the heating and tempering process is this: As the steel is heated, different crystal structures become stable at different temperatures. At the right temperature, and given the right combination of iron, carbon and other alloying ingredients, a certain form of crystal becomes stable (which was not stable at any other temperature). As the swordsmith maintains the piece of steel at that temperature, more and more such crystals grow within the body of the sword. At the right moment, when enough such crystals have grown, the sword is plunged into a cool liquid--usually water, but in the old days, they used blood to give a carburised coating to the sword.
Those desireable crystals have a structure called Face-Centered Cubic (FCC) (in steel). They grow quite fast given the right conditions, but if the material is quenched suddenly, there is not time for the atoms to re-arrange themselves into a new crystal structure, and the FCC crystal lattice gets squashed, eventually becoming the topologically simlar body centered cubic (BCC) arrangement (in steel) as the material cools. This leads to characteristic crystals of a long, thin shape. Such crystals are known as martensites, (whether they are in ferrous or in non-ferrous materials).
Steel is a bad example, because most materials tend to prefer the BCC crystal structures close to the melting point, (whereas steel likes to be FCC when hot) and the BCC crystals distort to the FCC when cold. Steel is the other way around.
Getting back to that sword, when the metal is first heated, it becomes a form of steel called Austenite. On quenching, almost all of that Austenite converts to martensite. The trouble is that martensite is also very brittle. With steel, there is usually a correlation--harder materials are more brittle, so the aim in the sword is to get the outer edge of the sword hard and (usually) brittle to take a good cutting edge, but the inner parts should be more softer and tougher, capable of absorbing the shock of a hard blow without breaking. This is why the sword is often given a carburised coating, which makes the outer layers harder. It is also one of the objectives of tempering, in which the temperature is raised to a few hundred degrees C well below melting point--and then rapidy cooled once more. In this process, some of the martensitic crystals are given time to change into other crystal structures, making the material a little less hard. This is a much more controlled process than the original production of martensitic structures, and allows the swordsmith to choose just how hard and how brittle the steel will eventually become.
Martens' discovery was the first time optical techniques had been used to identify crystal structures and make a significant contribution to the study of the physical properties of metals. Up to that time, people had tested hardness, and strength and so on with big machines, Martens was the first to simply look at a prepared surface under a microscpe. The martensitic patterns look best at around 1000 times magnification