Lawyers consistently screw up on this point. The attorney-client privilege extends only to communications and not to facts underlying communications. Here is a perfectly legitimate question: "When did you and your attorney meet?" Here is a perfectly inappropriate question: "What did you say?" There is a difference between these two questions, and over-protective lawyers often ignore it.
Existence of Attorney-Client Relationship
The existence of an attorney-client relationship is not privileged - it is a fact. Anyone telling you otherwise is wrong.
Identity of Client
Generally, a client's identity is also not privileged.
Purpose for Which Attorney is Retained
Attorney-client privilege does not extend to why the attorney was engaged.
Fee agreements are also not privileged.
Third-party witness statements are not privileged.