Technically, a corporate design is the official graphical design of a company's name and logo which is used across its marketing material. When a company has any new material made they want to be sure the new designs match the corporate branding - or are 'on brand' - this means that everything flows together and the company can be identified through the branding alone.
The most important thing to think about here, is that your brand stays with you. For MacDonald's, it could be the red and yellow colours that make people instantly think of them, or of late, it could be the Justin Timberlake whistle cover or just the words themselves - I'm loving it.
For corporate, b2b companies. There usually is no shop front, and generally speaking no tv advert. This means that the company's corporate web design is the first, and sometimes only place their customers will see.
However, with websites going mobile - it's important that you select a corporate web designer who is savvy enough to create an identity that fits across the corporate web site, the company's business card, and also looks good mobile.
Ideally, your corporate identity will look great on your website but also be adaptable in a way that creates unison across all of your corporate marketing and in house material.