Power
Not a lot of power. Graphic designers are basically artists who're commissioned to create art for someone else. The client tends to have strict deadlines and their own idea for their design. It absolutely doesn't matter how wrong you think the client is about a design.
For example, they may want a website for their company, which sells chicken poop to environmentally-friendly gardeners. They want the website to look "fun," "sophisticated," and "streamlined." They could send you a link to the website that they went to in the south of France as an example. It's your job to make them happy. Good luck making poop look sophisticated. Maybe you could attach a bow tie.
Working for an agency is slightly different. Agencies employ people to work with the client. The graphic design department may receive an email with a list of things that the client needs using their existing logo and brand. While there's not a lot of creativity involved in this process, there's considerably less guesswork when it comes to working with an established client.
If the design doesn't work, it's up to the graphic design team to make the appropriate changes. For large advertising agencies, graphic designers may work late into the night making changes so the client can review them in the morning. So much for settling in for that Mythbusters marathon you were so pumped about.