1) A proceeding conducted by an administrative or executive official that is similar to a court proceeding, e.g. a hearing. A court may review a decision arising from a quasi-judicial proceeding.

2) A judicial act performed by an official who is either not a judge or not acting in his or her capacity as a judge. 

Illustrative caselaw

See, e.g. Intel Corp. v. Advanced Micro Devices, Inc., 542 U.S. 241 (2004) and Humphrey's Ex'r v. United States, 295 U.S. 602 (1935)

See also