There are many more different kinds of brushes, but most fall under one of these six types.
Flat brushes are just what they sound like- flat and wide, and straight along the top. They can create a wide stroke when pulled in one direction, and a narrow stroke when turned and pulled along their edge. They are excellent for filling a lot of space quickly with paint, and are commonly used with oils and acrylics. A shorter flat brush is known as a bright.
Round brushes can be used with oils, acrylics and watercolors. They create a narrower stroke that can vary in width depending on the amount of pressure applies, and they are also great for creating dabs, dots and very small, precise strokes. Very long, very thin rounds are also called riggers.
Fan brushes are also known as blenders as their soft, fanned-out bristles are excellent for blending and feathering one layer of paint into another.
Filbert brushes are quite versatile, combining many of the features of flats and rounds to create a wide variety of stokes.
Angled shaders, also known as slanted or angular brushes are good for creating sweeping curves and calligraphic brush strokes, as well as fine-tuned and precise marks.
Mops look a lot like the powder brush in a makeup kit. These are round brushes preferred by watercolorists because of their ability to hold lots of fluid paint, making them perfect for washes.