And Deborah, a prophetess, the wife of Lapidoth, judged Israel at that time. And she lived under the palm tree of Deborah between Ramah and Bethel in mount Ephraim. And the sons of Israel came up to her for judgment.
(Judges 4:4-5)

prophetess
H5031
neb-ee-yaw'
Feminine of H5030; a prophetess or (generally) inspired woman; by implication a poetess; by association a prophet's wife: - prophetess.

judged
H8199
shaw-fat'
A primitive root; to judge, that is, pronounce sentence (for or against); by implication to vindicate or punish; by extension to govern; passively to litigate (literally or figuratively): - + avenge, X that condemn, contend, defend, execute (judgment), (be a) judge (-ment), X needs, plead, reason, rule.

judgement
H4941
mish-pawt'
From H8199; properly a verdict (favorable or unfavorable) pronounced judicially, especially a sentence or formal decree (human or (particularly) divine law, individual or collectively), including the act, the place, the suit, the crime, and the penalty; abstractly justice, including a particular right, or privilege (statutory or customary), or even a style: - + adversary, ceremony, charge, X crime, custom, desert, determination, discretion, disposing, due, fashion, form, to be judged, judgment, just (-ice, -ly), (manner of) law (-ful), manner, measure, (due) order, ordinance, right, sentence, usest, X worthy, + wrong.

Deborah, a prophetess, enjoyed a special relationship with God and served Him in a special capacity. He gave her the responsibility to judge both women and men. This seems in contradiction with

1Ti 2:12 But I do not allow a woman to teach, or to exercise authority over a man, but to be in silence.

Her primary role, which was mentioned first, was as a prophetess. Her authority as a judge seems to spring from that, as did her later role as a military leader. So her authority (civil and spiritual) was God-ordained. The specific example of her authority and the respect she commanded is here:

And she sent and called for Barak the son of Abinoam out of Kedesh in Naphtali, and said to him, Has not Jehovah, the God of Israel commanded, saying, Go and draw toward mount Tabor, and take with you ten thousand men of the sons of Naphtali and of the sons of Zebulun? (Judges 4:6)

And Barak said to her, If you will go with me, then I will go. But if you will not go with me, I will not go. (Judges 4:8)

She sent for Barak and he came - whether he perceived himself as subordinate to her position as a civil authority or in recognition that she was God's special servant and it was to his benefit to come. She clarifies that Jehovah is talking; this is not some whim of Deborah's to send him out to battle. But Barak's faith is weak. The word of God, delivered via Deborah, is not enough. He needs some hand-holding here, but he is at least willing to go. However, when we don't do what God wants when He directs us to do it, consequences are inevitable - the blessing of being the instrument used to enact God's will goes to someone else.

And she said, I will surely go with you. But the journey that you take shall not be for your honor, for Jehovah shall sell Sisera into the hand of a woman. And Deborah arose, and went with Barak to Kedesh. (Judges 4:9)

(Barnes) Mark the unhesitating faith and courage of Deborah, and the rebuke to Barak’s timidity, "the Lord shall sell Sisera into the hand of a woman" (Jael, Jdg_4:22). For a similar use of a weak instrument, that the excellency of the power might be of God, compare the history of Gideon and his 300, David and his sling, Shamgar and his ox-goad, Samson and the jawbone of the ass. (See 1Co_1:26, 1Co_1:31.)

You know the rest of the story - or if you don't go read Judges 4 and 5. I have been mulling over this for a week about Deborah's authoritative role in comparison with other edicts concerning the roles of men and women. Do I think women ought to be barefoot and pregnant and subservient to men? NO. But then that's not God's plan either. Women are not biblically restricted from having jobs or businesses, in fact the entrepreneurial spirit in women is specifically encouraged in several places in the bible. But where a woman enters into a marital contract and subsequently bears children, the bible insists that those obligations supercede all others except obligations to God. One thing I noticed is that Deborah apparently does not have children. Maybe that's part of the answer. Husbands take some maintenance, but a lot less than kids. As to the teaching and exercising spiritual authority over man, I need to continue to study this, and hope to post again on it after I conclude something worth writing about.