An affirmative case is one you bring yourself. You must be physically present in the United States, and you may apply regardless of how you arrived or what your current status is. You file Form I-589, Application for Asylum and for Withholding of Removal, with USCIS and you are interviewed by an asylum officer.
A defensive case is asylum raised as a defence against removal. You are in proceedings before an immigration judge, and the government is represented by an attorney from Immigration and Customs Enforcement. It is adversarial in a way the affirmative interview is not.
People generally land in the defensive process one of two ways: USCIS did not approve the affirmative application and referred the case to a judge, or they were placed in removal proceedings directly. A referral is not a denial in the ordinary sense — the judge conducts a de novo hearing, meaning a fresh hearing with a decision independent of what USCIS concluded.
If USCIS lacks jurisdiction, the asylum office may instead issue Form I-863, a Notice of Referral to Immigration Judge, for an asylum-only hearing.