Whether or not awards are important for you and your site really depends on your site.
For instance, this site is about design, and with it I'm offering all sorts of free goodies. I'm also offering my services as a web site designer. Because of this it is useful if I have my design work recognised by other designers on the Web. You can get hold of a good number of sites which make awards on http://www.award-it.com. It pays to spend some time looking up the categories of the different awards as not all will be appropriate for your site. When you get an award don't forget to use their banner as a hyperlink back to their site. It is the least you can do to repay the honour!

Finally, please don't lay claim to awards you have not earned. It's a betrayal of the trust on which most of the Internet Community relies.