You can't. The artboard won't print out, like if you had a circle shape on the artboard and print it on blue paper, the white artboard won't print out, just the circle on the artboard. The artboard is just a guide to let you know where to put your objects so you don't go outside of the area that will be printed.
You just use the eclipse tool to create the circle and work within that area, and when you go to print it out, then just the vector shapes and text and all will be what appears on the paper.
But if you're going to put it on the web, and want it to be just the shape of the logo, then you will have to save it for the web. Then you would go to File > Save for Web & Devices (it may just say save for web). Then that will bring up a dialog box. In it, you will be able to see your objects (the vector shapes you created) and they will be infront of a checkered grey and white box, that means everything behind the logo will be transparent, so you won't get a white box around it. The web supports gif.
Png is better with more colors and better with vector, but its not supported on all browser, so go with gif.