How can I show I know more than just haven taken courses in web development? Should I try to build some fancy project? If so, what kind of fancy projects would impress an employer? I know it should be user friendly but how can I make them impressed? How can I show I can do the job well and that they should hire me instead of someone who's experienced?

Make sure you understand the concepts of database programming in SQL as well, if that wasn't already covered in your courses. Almost all websites are based on SQL these days.
Go ahead and build the fancy project - you'll learn a lot in the process, and feel ready to get started with a job in the field when you're done with it. That alone will give you confidence. Make sure to follow good coding practices, such as thoroughly remarking your code, using descriptive variables, and so forth. I once built a simple registration system with email verification using SQL. Something like this relevant to real world business needs is good - a chess game might prove you're a genius programmer, but it doesn't show that you can fill the needs of a business the way a forum, shopping cart system, or other online system would.