Look at the rôle of women in the Plymouth Brethren:
"But I would have you know, that the head of every man is Christ; and the head of the woman is the man; and the head of Christ is God."
"1 The Head of every man is Christ – no equality. 2 The head of the woman is the man – equality and subjection. 3 The Head of Christ is God – equality, yet subjection."[42]
Thus most Brethren meetings reserve public leadership and teaching roles to men, based on 1 Timothy 2:11,12...:
"A woman should learn in quietness and full submission. I do not permit a woman to teach or to have authority over a man; she must be silent."
Also, 1 Corinthians 14:34,35 states, "Let your women keep silence in the churches: for it is not permitted unto them to speak; but they are commanded to be under obedience, as also saith the law. And if they will learn any thing, let them ask their husbands at home: for it is a shame for women to speak in the church." (The reason for this has to do with acknowledging Headship: Headship and the head covering are seen by many as inseparable since the head covering is intended to teach the meaning of headship. See below for information on the head covering).
From this, Brethren teaching traditionally (there are regional exceptions) outlines a system in which the men take the "vocal" and leadership roles and the women take supportive and "silent" roles. In practical terms, what is traditionally seen is that the men are fully responsible for all preaching, teaching and leading of worship. Therefore, in most Brethren groups women will be heard to sing the hymns along with the group, but their voices will not otherwise be heard during the meeting. Often the men are, practically speaking, the only ones involved fully and vocally in all discussions leading up to administrative decision making as well. Within Exclusive groups in particular, matters up for debate may be discussed at special meetings attended solely by adult males called, in some groups, "Brothers Meetings".
The Head Covering:
As to the reason behind women covering their heads at meetings in some groups, 1 Corinthians 11:5,6 says:
"But every woman that prays or prophesies with her head uncovered dishonours her head: for that is even all one as if she were shaven. For if the woman be not covered, let her also be shorn: but if it be a shame for a woman to be shorn or shaven, let her be covered."
The verse indicates that when a woman speaks in church (i.e. prophesies) she is supposed to have her head covered. The Brethren generally leave this part out in the meetings and have the women be silent while wearing head coverings.
For this reason, some meetings will be characterized by the women wearing head coverings ("loaners" in some assemblies are available at the back for women who have come without a covering). Head coverings typically take the form of a tam, beret or similar hat which can be more aptly described as a "head topping," rather than as covering the head in any real way. Sisters in Exclusive ('Jimite') gatherings quite commonly wear a headscarf or "mantilla" (a lace/doily-like Spanish veil) on their heads. It is a fairly common misconception that Exclusive women characteristically wear a shawl over their heads, though no doubt some women have sometimes resorted to this.
While that is an overly-simplified view of the head covering, understanding the purpose for the head covering comes from looking at 1 Corinthians 11:3&4, which says:
"3But I would have you know, that the head of every man is Christ; and the head of the woman is the man; and the head of Christ is God. 4 Every man praying or prophesying, having his head covered, dishonoreth his head."