The first piece of advice I offer about choosing is bow is to try a number of them. There is no "best bow" for everyone and every instrument. A bow that makes one violin sing, for example, might deaden the sound another. Likewise, a bow that is supple and light may be perfect for some players, but probably won't work for someone who plays with a lot of pressure on the bow. It's not a matter of which is better; it's just that they're different. Your goal is to find a bow that draws the best sound from your instrument, while making your work as a player as easy as possible.
When trying a bow, I like first to listen to the sound that it draws from an instrument. This is purely a matter of taste, so what sounds best to you is what's most important. Draw long even strokes on each string. The affect a bow has on sound is determined by the interacting resonances of the bow and the instrument, and by the dampening effect of the bow, and is very difficult to predict. If you don't like the sound of a bow, set it aside and try another. After you've narrowed down your choices based on sound to two or three bows, then see how they feel and how they perform various strokes. Again, take long slow strokes. If the bow has weak spots in the stick, or unevenness in the curve of the stick, you'll often find that it chatters at the same spot with each stroke. Try off the string strokes; you may have to search a bit for the "sweet spot", where the bow bounces best, which can be in different places on different bows. Try trills; do the individual notes speak clearly. Does the weight of the bow feel right to you? Don't worry too much about the bow's weight in grams, since balance can greatly affect the perceived weight. When you've found one or two bows that seem to work for you, it's then important to play them for extended periods in the settings in which you'll ultimately be using them. Give yourself time to get used to a bow, and to learn what it can and can't do.
If you would like to evaluate bows in a more technical way, there are a number of things to look at:
Materials: Fine bows are almost always made from pernambuco, a very strong, dense wood from eastern Brazil. Its strength and resonant qualities are what set it apart from other materials. A bow made from other woods would have to be very heavy in order have acceptable strength. As for the resonant qualities, this is a very complex matter. Suffice it to say that just as spruce is far and away the best wood for instrument soundboards, the same can be said of pernambuco for bows. Unfortunately for us today pernambuco is a rare and endangered wood. It grows in the rainforests of Brazil, and is subject to the pressures of development and the struggles of many of the people there to survive. The wood being used now is from existing stocks, as the Brazilian government is not allowing any pernambuco to be harvested. Many bow makers are experimenting with other woods, trying to find an acceptable substitute for pernambuco. So far these efforts have met with only limited success.
The grain of the wood should run fairly straight through the stick. If the grain has "run out", meaning that it is oriented diagonally across the stick, it is weaker at that point than straight grained wood would be. A little run out in the thicker two thirds of the stick is acceptable. However, it's important that the grain runs straight in the upper third of the bow near the tip. Because this area is thinner and more delicate, it is susceptible to damage. Straight grained wood will lessen the possibility of a broken stick or tip.
Less expensive bows are often advertised as being made of "Brazilwood". This has come to be a generic term for just about any wood other than pernambuco. While these bows may be fine for beginning students, they won't perform as required for more advanced playing.
Frogs are generally made of ebony, with fittings of nickel silver, Sterling silver or gold. So called "nickel silver" actually has no silver content, and is used only on inexpensive bows. Sterling silver, or other silver alloys are the standard, and are used on bows ranging from higher grade student bows to very fine bows. The use of gold on frogs is, for the most part, cosmetic. Bow makers save their best sticks, and strive to do their best work on gold mounted bows. The shell used for the slide and eyes in the frog and button won't have an affect on the playing qualities of a bow, but some types wear better than others. The salt and acids in perspiration can deteriorate shell. White mother-of-pearl from oysters wears very little over time compared to more colorful abalone. (This wear can be minimized by wiping the frog with a clean cloth after each use.)
The mechanics of the bow: The frog should fit well on the stick, without too much side-to-side movement. There has to be a little play here, so that the frog can slide freely back and forth. When the hair is under tension, the back of the frog may lift from a stick just a little, but the front should show no gaps and the frog should feel secure on the stick. Feel the sides of the stick and the frog where they meet. They should be even with each other; the frog should not be wider or narrower than the stick at this junction. The button should turn freely, and have only minimal play. Push side-to-side on the button; you should feel only the slightest movement.
The camber or curve of the stick: When you tighten up the bow, the camber should be a smooth even curve with no kinks or flat spots. Hold the bow right up to your eye, looking from the button towards the tip, and sight down the stick. If you over-tighten the bow a bit any flaws in the camber will be exaggerated and easier to see. You should also check to see that the bow is fairly straight from side to side. As you sight down the bow from button to tip, a little curve to the left is acceptable for violin and viola bows. Any curve to the right is definitely not OK. This is because when you're holding the bow in the typical playing position it is leaning away from you, or to the right when viewed from the frog end of the stick. If the stick is already curved in that direction, the tendency for the stick to hit the strings will increase. Because of the way cello and bass bows are held, the opposite is true; a little curve to the right is acceptable.
The weight should be in the range of 58 - 64 grams for violin bows, 65 - 75 grams for viola bows, and 78 to 85 grams for cello bows. These figures represent the extremes. Weights of 60 grams for violin, 70 grams for viola and 80 grams for cello are considered about the median. Bass bows are much less standardized, and can typically vary in weight from 125 to 160 grams. I often advise players looking for a bow not to be fixated on having a bow of a certain specific weight. The balance of the bow is also an important factor, and can affect the feel of a bow at least as much, if not more so than the weight. A simple demonstration can make this clear: Take two bows that weigh exactly the same and have the same balance point. Tape a nickel to the tip of one bow, and a nickel to the frog of the other. The bow with the nickel at the tip will feel much heavier than one with the nickel at the frog, because of the leverage of the weight so far from your hand compared with the direct pull of the nickel on the frog. It's not necessary to know the measurement of the balance point of a bow when evaluating it. How it feels in your hand is what's most important. If you are interested in measuring the balance point, this is a method commonly used by bow makers: Adjust the frog to the forward-most position, so that it touches or is at least close to the thumb leather. Screw in the button until it just touches the end of the stick. Then balance the bow on your finger or a pencil, note where that point is, and measure from there to the end of the stick where it touches the button. This measurement should be in the range of 8 ½" to 10", except for bass bows which generally have a balance point a bit closer to the frog. The closer the balance point is to the tip, the heavier the bow will feel. Conversely, the closer the balance point is to the frog, the lighter the bow will feel.
The strength or stiffness of the stick: When a bow is tightened to normal tension, the stick should not hit the strings when normal playing pressure is applied. What's normal bow pressure can vary a lot from player to player, and good bows can range from quite flexible to very strong. If you have to over-tighten a very flexible bow in order to keep the stick from hitting the strings, the stick can get "squirrelly" and tend to flex side to side when pressure is applied. A bow can also be too stiff, which may hinder the way it bounces and possibly the sound it generates. I've found that many good players often use very flexible bows. Rather than using a lot a pressure to draw a big sound, they use more bow speed, or the proper balance of speed and pressure. That being said, I also know some good players who like very strong sticks. Bow preference is a very individual thing.
This may seem like a lot of factors to consider. It's important to keep in mind that playing music is not like operating a machine. Trust you instincts. What feels best, sounds best and makes music making the most enjoyable is what's important.