A CDI is an individual who is deaf or hard of hearing and has been certified as an interpreter by RID.

Why? Typically equipped with excellent communication skills, a CDI may also have special training or experience with gesture, mime, props, and other tools. They can also provide extensive expertise on deafness and the deaf community.

When? CDIs may be needed when a deaf client's method of communication is so unique that it cannot be adequately accessed by hearing interpreters. Situations might include individuals who use non-standard "home signs," a regional or foreign sign language, or who are deaf-blind. A CDI might also be useful in an area that is not adequately arranged for visual communication, in which case they are typically positioned so they can be seen by two parties who cannot see each other; in this case the CDI repeats or mirrors each statement.

How? CDIs often work in teams with certified hearing interpreters. In some cases they will transmit the message from the deaf client to the hearing interpreter, who then passes it along to the hearing client; in other situations a deaf and hearing interpreter will work together to arrive at the best interpretation. When a deaf-blind person attends a presentation given in sign language, a CDI will take the signed message and adapt it at close range for the client.

Source: http://www.rid.org/cdi.html