Is a patient's family always asked whether they want to donate the patient's organs?

No. If the donor is active on the kidney registry, the family does not need to be asked. Otherwise, federal law requires asking the family except under medical circumstances under which the patient is clearly not a potential donor.
Yes. Permission from a patient's next of kin is required before deceased organ donation. However, there may be differences of opinion among spouses, families, or relatives. It is always best for an individual to express his/her wishes while alive, and confirm in a written document or driver's license.