A symbol is a representation of something.

This representation can be done using any technique, could be graphic, character, sound etc.

For e.g.: when we write CAT, it's not the actual animal but just a charter set that symbolizes the animal. This could also have been done by drawing the cat, which would again be a symbol for the cat.
Summarizing it again, Symbol = Representation

When we talk of branding, it about attaching a sense of ownership. It’s a method of putting a symbol so that the product can be traced to its owner.

Also the symbol used is made such that it is easy for the viewer to understand, recognize and relate to the owner.

The graphical symbol used in case of a brand is called a logo.
The associated text used is called the logotype and together they form the trade mark for the brand.