Here's a list of standard types you'll find when shopping for brushes. As an artist who's worked in many mediums, I own them all. However, my favorites are flats and rounds. The art store is a wonderland for me!
Basic Brush Types
Angle brush - A favorite in the Flat family for its abilities to handle tight shading and lines. Use it to attack smaller areas.
Chisel blender - This is also called a Flat, but it features shorter bristles. It can't hold as much paint or water, but it's a great tool for blending.
Filbert brush - Also known as a cat's tongue for its rounded end shaping. You'll typically get softer strokes and it won't leave "raking" behind, which flats tend to do. Also, in various sizes, it makes creating flower petals so easy.
Flat brush - This will be the most versatile style in your artist's box. In a variety of sizes and types of bristles, it handles just about everything you need. At the tip, it can produce finer lines and then across wider expanses on its side.
Fan brush - Used for texturing, can create swirls, wood graining, etc. Keep it dry and rake it across wet paints or see what you can do with it when it's wet.
Mop brush - Also used for blending and a nice Filbert will substitute.
Rigger or Liner brush - When you need an overlay of finer lines, this is the brush you'll need. It doesn't hold up under pressure - just a light touch will do. With longer bristles, it's available in multiple sizes.
Round brush - A favorite for watercolors, but with its pointed tip, it also produces finer lines in oils and acrylics. Mastering control can be difficult, but it's an essential for smaller works or any piece that requires a bit of a technical touch.