There are multiple steps to designing or redesigning a website. Depending on the audience, e.g. business-to-consumer (B2C), or business-to-business (B2B), or intranet, the process will vary slightly, but the steps are still the same. In a redesign, you are going into the process to improve the current website, so you have existing data that you can work with to improve the outcome. In a new design, you are starting from a clean canvas, so it is imperative you get it right the first time and know who your audience is.
You'll want to perform user interviews, create personas, use-case stories, and process workflows before you even start creating wireframes and mockups. It's easier to make changes on paper than it is to go in the backend and start making code changes.
Below are samples of my web design and user interface/user experience (UI/UX) work. Enjoy!