The bow used for string instruments operates on the principle of "grip-slip". When the bow hair is drawn across a string, it grips it and pulls it to the side. As the bow is moved, the tension grows until the resistance of the string overcomes the grip of the bow hair, causing the string to snap back while the bow hair slips. The energy travels along the string to its stopping point, and then back on its other side until reaching the bow hair again, at which point it grips the string again and the cycle is repeated.
Bows are constructed of four parts: the stick, the frog, the hair and the screw. The stick carries the hair and provides the elastic function of the bow. The hair is attached at one end to the tip of the stick, and its other end is attached to the frog. The frog is a piece of hardwood (usually ebony), bone, ivory, or some other hard material which is used to space the bow hair from the bow stick. In antique bows, the hair was attached to the stick instead of to the frog, and the frog would be clipped-in between the hair and the stick. In modern bows, the frog rides on a groove in the underside of the stick. The screw changes the frog's position in the groove, and thus tightens or loosens the hair.
Bow hair is made of horse-tail hairs. The most sought after hair comes from Siberia, Mongolia and China. There are two types of hair: white hair and black hair. White horse hair is finer than black horse hair, which tends to be coarse. Some contrabass and cello players prefer black hair because it grips better. Others prefer a mixture of white and black hair, which is called "salt'n'pepper". The bow hair needs some sticky rosin applied to it in order to adhere to the string, but care should be taken not to apply too much rosin as it may lead to a choked sound. The hair is extremely sensitive to dirt and one must refrain from touching it. If it becomes dirty, it can be cleaned with denatured alcohol (care should be taken not to apply the alcohol to the stick, as it will ruin the finish and leave stains). With use, the bow hair eventually becomes too stretched and can't anymore be brought to playing tension. This is fixed by rehairing the bow, which should be done by a professional bow rehairer.
Although cheap bows are available which are made from plastic or fiberglass, better quality bows are made from different types of hardwood (usually brazilwood or lower-grade pernambuco), and master bows are usually made from select pernambuco or snakewood. Snakewood is also popular for the construction of baroque bows. The most important requirements for bow stick wood are density and uniformity of grain. Recently some makers began making bow sticks from carbon-fiber, trying to imitate the stiffness of high quality pernambuco.
The bow stick should be heavy enough to be able to grip the strings, and at the same time not too stiff, in order not to stifle vibrations. Another factor in the construction of the stick is balance. Some bows might have a very heavy tip, which might be useful for obtaining a strong sound, but makes the bow a bit clumsy. Other bows have a lighter tip, which will make them more agile, but will demand use of additional weight in the bowing arm. Bow sticks are also made in different shapes. Some models are curved inwards (as in most modern bows), some (usually the renaissance and baroque models) outwards, some are almost straight.
Different types of instruments have different types of bows in varying dimensions. For example, a viola da gamba bow is very different than a violin bow. And for each instrument there might be different models, for example the French (Bottesini) and German (Dragonetti) models for the contrabass. In addition, bows are constructed with different stiffnesses and weights. The choice of a bow is a matter of taste as well as musical requirements. For example, a baroque violin player will need a different bow than a violinist who plays modern repertoire. One player may prefer a stiff and light bow, while another would like a heavy bow with more elasticity.
The bow is considered by string players to be the 'soul' of the sound. Especially in historically-aware performance of renaissance and baroque music, musical expression is controlled by the bow (in contrast to the use of vibrato as the main expressive device in modern string playing). The modern violin Tourte bow was designed to obtain an evenly sustained sound, while the baroque violin bow is designed to produce a 'bell-tone' (clear attack and gradual decay). Some bows, called spiccato bows, are specifically designed to bounce off the string for clear articulation in the execution of fast stacatto passages.
For an excellent discussion of bows and their construction visit http://www.xs4all.nl/~bowmaker