Every designer has his own way of working. I present you my design process.
First I have to understand what the customer wants me to do, and his priorities. It is essential to know exactly the goal of the project to realize it.
The next step is to search some inspirations around the theme of the project. It also allow me to see what is already done and so it give me an idea of what is "good" or not.
Then I can start drawing some sketches of elements (logos, icons...) or any ideas relating to the request. I draw some interfaces too, at least the type pages.
This step is one of the most important.
I regroup my good ideas from the sketches and I design the interfaces. I strive to make them easy to use and beautiful.
Finally, I control if it all works in any browsers or resolutions. And I check all the pages tracking down any missing links or any errors (spelling mistakes...)
I have forgotten to tell you one thing I do during all this steps... Listen to music!