A posek answers a question of Jewish law, a she'elah, with a ruling, a teshuvah, and the ruling carries weight because of who issued it and how, not because any office ordained the person to issue it. There is no ordination that makes someone a posek. The status is earned by recognized mastery of the halakhic literature and accepted by a community that chooses to follow the ruling, which is why a posek's authority is local and reputational rather than institutional in the way a bishop's office is institutional.
The method is built on precedent. A posek does not decide a new question from first principles but works outward from the existing codes, chiefly Joseph Karo's sixteenth century Shulchan Aruch and its major commentaries, through the accumulated responsa literature of earlier poskim addressing comparable cases, toward the specific facts of the question at hand. Two poskim working from the same sources can reach different conclusions, since applying precedent to a new case is itself an act of judgment, and Jewish legal history is full of named, respected disagreements between major poskim rather than a single settled answer to every question.
A posek's reach is also bounded by specialization. Some poskim are recognized authorities on the laws of Shabbat, others on family law, others on the dietary laws that keep the kashrut system running, and a community or a mashgiach with a hard case will seek out the posek whose recognized expertise actually matches the question, rather than treating any one rabbi's ruling as automatically authoritative outside his area.