The term "e-business", meaning to conduct business through the usage of Internet, was originally coined by IBM back in 1997 as part of a marketing campaign.
This definition is potentially misleading and is often confused with the term "e-commerce", which specifically refers to buying and selling products and services online (i.e. a component part of e-business).
UK Online for Business, the DTI's e-business initiative, offers the following superb explanation of what e-business is and why it's essential:

E-business is about more than just technology. It's about thinking what your customers need and using internet and other information and communication technologies and applications to increase business performance and success.
Internet technologies can be used to build relationships with customers, automate ordering, allow secure payment, speed processes and can significantly reduce costs. Whatever you deliver, whether its goods or services to customers, or parts to a factory, it will need to be coordinated using technology that makes the most of streamlined and tightly integrated business processes.

E-business.... describes a.... degree of integration of communications technologies with business processes and management practices, often conducted via the internet. It has implications that are inward as well as outward facing.
The key to e-business success is understanding how your customers work as well as adapting the running of your business.
Think first about the business, and what you need to achieve then consider the technology. The business should not be driven by technology but understand how technology can help your business and address your real business needs.

Helping a clients achieve this understanding in order to improve business performance is at the heart of what s e-business management consultants do.