What do we mean by a "Graphic Design".
If you already have a company logo then you will want that included in your web site. Perhaps you utilized a printer to create one for your business cards. Perhaps you created one yourself when creating a brochure or flyer.
You might hire a professional graphic designer to create your logo. Graphic Designers charge anywhere from $35 to $70 per hour to create graphics. They will use software such as Paint Shop, Photo Shop, or Fire Works. There are dozens of products available. Your logo might include your company name, your slogan, or type of business. The graphic designer will try different backgrounds and fonts to get the desired effect.
You could try to design your own logo. If you have a word processing program like Word, you could import some clip art and then try out different fonts for the wording of the logo. You could purchase software like Paint Shop for less than $100 to create some cool visual effects.
Once you have your logo designed you will need to forward it to your web designer to include in your web site. It will need to be saved in either a JPEG or GIF format. This might be different from the format used by your printer or graphic designer.
You could have your web designer create your logo. They will also have the necessary software to create one. I would suggest being explicit on what kind of look you want and what wording should be included.
Your logo should not be too large or contain too much information. It will be included on every web page. If it is large it might cause the downloading of the page to take too long. Also if it is too large it might take up a lot of room on the page that is needed for the content of the web page.
Beyond your logo, you might want other graphics on your site. This could be navigation buttons or small graphics in the corner of a page or a background texture. In general you can avoid using these types of graphics. Your web pages will be most effective with a solid light color background. Your navigation buttons can be created with simple borders and colors. You can leave the corners of your pages blank. By eliminating these unnecessary graphic elements you can save yourself some money and present a cleaner, quicker web site.
Again if you do want a specific element included in the background or in a specific area of your pages you could hire a graphic designer, design it your self or pay your web designer to do it.