When working in a "virtual" or computer-to-computer environment, some form of unique identification must accompany any transaction that is sent to the card processor. This, again, is how the processor knows where the transaction originated and that it is being executed for your account. You will receive other identification information from the card processor when you set up your account. These "client numbers" help to identify you to the processor, but are not the same as the terminal identification number.
The reason for having both is that the client identification is usually unique to a customer and credit card type - hence, you may receive multiple client numbers, one for each credit card type that you wish to accept. At the same time, you may have more than one card terminal - one at your front desk, one in the back office, and the new virtual terminal you are creating for Internet processing. Each of these will also have a unique identifier.
Part of the process of "signing up" with a partner processing agency will be to provide them the terminal identifier for your "virtual" terminal that was assigned by the card processor. It is important to request and keep this identifier with your processor account information.