Lets start by defining design. In short, it is the preliminary sketch or plans for a work to be executed.
So that makes a designer someone who devises or executes designs.
Now i'll stop reciting the dictionary, but looking at the above definitions it's pretty obvious that a designer has a lot of scope. Let us assume that the majority of designers design solutions to human needs (commercial, industrial, etc). The most widely known and/or recognised desgners include:
    - Costume Designers
    - Digital Media Designers
    - Fashion Designers
    - Graphic Designers
    - Industrial Designers
    - Interior Designers
    - Jewellery Designers
    - Textile Designers
There are many more designers out there who might specialise in Film and Set Design, etc. It is a void, an expanding industry, a constantly evolving concept. No matter what you personally are into, there is an area or niche in the industry that will suit you.
I have found this website for the Design Institute of Australia which has some great information about Design. Be sure to check it out.
http://www.dia.org.au/index.cfm?id=3