Unlike art, design always has a purpose. In web design, you always have goals to achieve. An e-commerce site should sell stuff. A marketing site should gather leads or communicate a message. A web-based application should allow people to carry out certain tasks.
Your goals define the problem you need to solve. Design is the process you follow to create the best possible solution to that problem. So web design does not start with graphics. It starts with understanding the challenge. Who am I communicating with?