What are the different type of paint brush bristles?
Natural vs synthetic

I did a bit of research into this and found that there are two main kinds of paintbrush bristles: natural (usually made from animal hair) and synthetic (usually made from some combination of nylon and polyester). Each of these has its own advantages and disadvantages according to what kind of paint you're using. The TL;DR is that natural bristles are better for oil-based paint while synthetic bristles are better for water-based or latex paint. For more details, see this page. As a bonus, it also gives some information on the shape as well as the material of paintbrush bristles:


  Bristle Shape: Most paintbrushes available today are square-cut brushes. They're perfect for holding and laying paint onto virtually any surface. However, square-cut brushes don't provide as much control when painting into corners, up to adjacent surfaces, or along narrow edges or surfaces. For more precise control, use a sash brush, which has its bristles cut at a slight angle. Sash brushes are particularly well-suited for cutting in around the perimeter of a room.
  
  Bristle Tips: Better quality brushes have bristles with flagged, or split, ends. Flagged bristles hold more paint and spread paint more smoothly. Some brushes, especially sash brushes, have tipped ends, which should not be confused with flagged ends. Tipped brushes come to a point; they're not cut flat and straight, as is a standard brush.
  
  Professional painters favor tipped brushes because they provide greater control and allow you to apply paint more precisely.


More details on different types

This list goes into more detail on the usages of synthetic and various types of natural bristles, although it doesn't distinguish between different types of synthetic material:


  
  Badger
  
  For blending oil paint on canvas, Badger Hair is an age-old tradtion. It comes from various parts of the world and is more readily available than most animal hair, although the quality varies greatly. Badger hair is thickest at the point, and relatively thin at the root, so it has a distinctive "bushy" appearance.
  
  MEDIA: oil
  Camel Hair
  
  Camel Hair does not come from camels at all. It is found in watercolor and lettering brushes and usually is made of squirrel, goat, ox, pony or a blend of several hairs, depending on the desired softness and intended cost of the brush.
  
  MEDIA: lettering, tempera, watercolor
  Hog Bristle
  
  Hog Bristle is obtained from hogs in several parts of the world, the most sought after coming from China. Bristle is unlike any other natural filler in that it forms a V-shaped split or "flag" at the tip and tends to have a natural curve. A brush with "interlocked" bristles, with the curves formed inward to the ferrule, has a natural resistance to fraying and spreads medium to thick paints smoothly and evenly. A selection of pure hog bristle brushes is recommended for oil and acrylic painting, and is a far less expensive alternative to good-quality softer hairs.
  
  MEDIA: acrylic, oil
  Kevrin/Mongoose Hair
  
  Kevrin/Mongoose Hair is strong, resilient, and makes a good long-wearing, medium to professional quality brush for oil and acrylic painting.
  
  MEDIA: acrylic, oil
  Kolinsky Sable
  
  Kolinsky Sable is not really from a sable at all, but comes from the tail of a species of mink that is a member of the weasel family found in Siberia and northeastern China. It is generally conceded to be the best material for oil and watercolor brushes due to its strength, spring and ability to retain its shape ("snap"). It holds a very fine point or edge. This is considered a professional grade of hair, and if properly cared for, Kolinsky will last for many years.
  
  MEDIA, oil, watercolor
  Ox Hair
  
  The best quality comes from the ears of cattle or oxen. The Ox Hair has a very strong body with silken texture, is very resilient, has good "snap", but lacks a fine tip. Therefore, it is most useful in medium gradewash brushes, or flat shaped brushes. Frequently, ox hair is blended with other natural hair to increase the resiliency of a brush.
  
  MEDIA: lettering, watercolor
  Pony Hair
  
  Pony Hair is soft but strong, from mature animals at least 2 years of age. It is primarily used for scholastic grade brushes, but often blended with other hairs for inexpensive watercolor and touch-up brushes.
  
  MEDIA: acrylic, scholastic, tempera, watercolor
  Red Sable
  
  Red Sable is obtained from any member of the weasel family with "red" hair, not at all from the animal known as the sable. It is found in a variety of brush styles for many varied mediums, with quality and characteristics varying greatly. A good quality pure Red Sable is a good alternative to the more expensive Kolinsky, with similar performance and durability. Often, weasel hair is blended with ox hair to make a more economical brush, but the fine point is sacrificed.
  
  MEDIA: oil, watercolor
  Sabeline
  
  Sabeline is actually select, light-colored ox hair dyed to resembled red sable. Lettering and watercolor brushes often use Sabeline mixed with Sable to lower the cost of a brush.
  
  MEDIA: lettering, watercolor
  Squirrel
  
  Gray Squirrel (Talayoutky), most highly in demand for lettering brushes and quills, is native to Russia and nearly always fell in short supply. Brown squirrel (Kazan) is more readily available, and is used mainly for medium quality and scholastic watercolor brushes. A very fine, thin hair, taken from squirrel tails, it points as well as Kolinsky, but has very little "snap" because the hair is not very resilient. It works best with liquid paints and inks.
  
  MEDIA: lettering, watercolor
  Synthetic
  
  Synthetics are man-made of either nylon or polyester filaments. They can be tapered, tipped, flagged, abraded or etched to increase color carrying ability. Often, synthetic filaments are dyed and baked to make them softer and more absorbent. The common name for this filament is "Taklon". Advantages of synthetic brushes are: 1) They are less prone to damage from solvents, insects or paints. 2) They are easier to keep clean than animal hair brushes because the filaments don't have animal scale structures to trap paint. 3) They are less prone to breakage and are durable on many different surfaces. 4) They are better suited for painting with acrylics because a synthetic filament will withstand the caustic nature of acrylic paints with less damage.
  
  MEDIA: all
  


Finally, this list goes into even greater detail on a huge variety of types of bristle, falling into four main families: animal hair, vegetable fibre, synthetic, and wire brush. Within each of these families, several different materials are described and compared. It's too long to copy it all here, but should be useful as further reading if the above isn't detailed enough.