Being able to accept feedback (or criticism, if you must) is an important part of a Graphic Designer's skill set. As an instructor and trainer I've seen people who can accept feedback always growing and improving and doing some pretty incredible things. If you're interested in improving your design and also making clients happy, one of the best things that you can do is work on accepting feedback.
The first step to accepting feedback is considering the source. If you're only showing your work to close friends who say nothing but, "Wow, that is so cool!", it will feel good, but it won't help you to grow and improve. Of course, you don't want to show your work to someone who just says, "I don't like it", that won't help either. Find someone whose opinion you respect and be prepared to change your work. They may be right. This is not an argument contest. If you are good at arguing, I suggest that you become a lawyer, not a designer.
Accepting feedback from clients is another consideration. Sure, it would be nice if they accepted your first designs, but they probably won't. If you are a good designer and a creative person, you can come up with lots of other designs. If you can't, I'm sorry, but you probably shouldn't be calling yourself a designer.
Technical feedback is especially crucial for web design. If someone tells you that something isn't working or displaying properly on a web page, that is your cue to go fix it, not tell them to view it in Firefox or at a particular screen resolution.
Good feedback is precious. Seek it out. Use it.