Vav is a gate to a host of possibilities

Vav is a hook. It's what keeps the pieces of the tabernacle together. When I was dabbling in grammar for my own learning last season, I didn't attempt to summarize this most important enclitic Hebrew consonant. Vav attaches itself to all sorts of things. It's a prefix with a great variety of uses and a suffix also. This is where its hook nature thrives. There are other hooks in the Hebrew Scripture: fishing hooks, pruning hooks, and grappling hooks. Vav as hook occurs in Exodus only in 13 verses. E.g.

וְהָאֲדָנִ֣ים לָֽעַמֻּדִים֮ נְחֹשֶׁת֒ וָוֵ֨י הָֽעַמּוּדִ֜ים וַחֲשׁוּקֵיהֶם֙ כֶּ֔סֶף וְצִפּ֥וּי רָאשֵׁיהֶ֖ם כָּ֑סֶף וְהֵם֙ מְחֻשָּׁקִ֣ים כֶּ֔סֶף כֹּ֖ל עַמֻּדֵ֥י הֶחָצֵֽר		17 And the sockets for the pillars, brass. The hooks of the pillars and their attachments, silver, and the overlaying of their tops, silver. And they were attached with silver, all the pillars of the court.
iz vhadnim lymudim nkowt vvi hymudim vkwuqihm csf vxipui rawihm csf vhm mkuwqim csf col ymudi hkxr	33 15	vh/adn\im l/ymd\im nkw\t vv\i h/ymvd\im v/kwvq\ihm csp v/xp\vi raw\ihm csp v/hm m/kwq\im csp cl ymd\i h/kxr

And the music:

Exodus 38:17

BDB describes vav as a conjunction and its use in the verbal system when it is a prefix for the verb form. The opening paragraph indicates that it is not "a merely copulative conj., but that it possessed a demonstrative force... ו is used very freely and widely in Heb., but also with much delicacy, to express relations and shades of meaning which Western languages would usually indicate by distinct particles... thus in AV, RV, words like or, then, but, notwithstanding, howbeit, so, thus, therefore, that, constantly appear, where the Heb. has simply ְו ."

Structurally BDB has five sections in the 6 pages.

1. And, connecting both words and sentences

2. The ו consecutive 

3. To express an intention, that or so that 

4. It expresses often an informal inference, or consequence, so, then

5. ו introduces the predicate or apodosis

BDB begins with a verse illustrating 'thus and so' or similar construction.

וַיִּ֤פֶן כֹּה֙ וָכֹ֔ה וַיַּ֖רְא כִּ֣י אֵ֣ין אִ֑ישׁ וַיַּךְ֙ אֶת־הַמִּצְרִ֔י וַֽיִּטְמְנֵ֖הוּ בַּחֽוֹל		12 And he faced thus and thus, and saw that there was no person, and he struck the Egyptian and buried him in the sand.
ib viipn ch vch vira ci ain aiw viç at-hmxri vi'tmnhu bkol	11 13	vi/pn ch v/ch vi/ra ci ain aiw v/ic at hm/xr\i vi/'tmn\hv b/kvl

This verse illustrates five prefixed vavs (red), the first, third, fourth, and fifth all introduce the narrative form of the verb (type 2 above) and are translated in the past tense rather than the imperfect. The third might have been rendered so based on the prior phrase.

Vav is also a pronoun - third person masculine singular, e.g. lv (to or for him / it etc) pronounced as lo and written as lo in SimHebrew (so distinguishing it from the negative la pronounced the same as lo), and vav as suffix plays an important role in verb forms., such as the first, second, and third person plural. 

These uses can be confusing. Here's an example of the confusion of not (la pronounced lo) and his (lv also pronounced lo - this is the preposition l with a pronoun suffix his = belonging to or for or of or to him) .

Psalm 100:3

דְּע֗וּ כִּֽי יְהוָה֮ ה֤וּא אֱלֹ֫הִ֥ים הֽוּא־עָ֭שָׂנוּ וְל֣וֹ אֲנַ֑חְנוּ עַ֝מּ֗וֹ וְצֹ֣אן מַרְעִיתֽוֹ		3 Know, for Yahweh he is God. He, he made us, and his we are, his people and the sheep of his pasture.
g dyu ci ihvh hua alohim hua-ywnu vlo anknu ymo vxan mryito	9 9 7	dy\v ci ihvh hva alh\im hva yw\nv v/l\v anknv ym\v v/xan m/ry\itv

The first vav, a suffixed u is the imperative, second person masculine plural, the second is the verb to make with a pronoun suffix us, nu, the third is the same nu but it is part of the standalone form of the first person plural pronoun. The third, I am reading what is written: lv = lo and not what is read, la - lamed-alpha לֹא.

You may think of this verse as requiring the phrase 'not we ourselves'. That requires reading lv as la. It is given in the manuscripts as what is to be read, but I have read diverse opinions as to the right sense. I think of this verse as reflecting the Song of Songs -- my Beloved is mine and I am his.

Song 2:16 

At some distant ancient time, I was working on this section to set it for woodwinds and voices -- looks like I set only the first poem to 1:1 to 2:7. If you are interested, there is a mechanical version here.

דּוֹדִ֥י לִי֙ וַאֲנִ֣י ל֔וֹ הָרֹעֶ֖ה בַּשּׁוֹשַׁנִּֽים		16 My beloved is mine, and I am his; he grazes among the lilies.
'tz dodi li vani lo hroyh bwownim	14	dvd\i l\i v/ani l\v h/ryh b/wvwn\im

BDB also mentions conjunctive and disjunctive accents. The music includes rhythm and might well ornament grammar. When I was first given the music, the notions of conjunctive and disjunctive as well as continuous dichotomy gradually disappeared from my vocabulary. They are both unnecessary concepts, vaguely defined and positively misleading. The music will never mislead you and it is defined in its design. This is the design of the whole tabernacle, temple, and upholding its fortification -- perhaps we could look at that verse Ezekiel 43:11 and its 36 vavs.

We could pick almost any verse and find a vav in it. Of the 305 thousand odd words, over a third of them have vav as part of the prefix, and over 25% if them have a vav as part of the suffix. How could we summarize the uses of vav among these many examples.

Vav is represented in SimHebrew as v or o or u depending on various contexts (see the highlights below).

Ezekiel 43:11

וְאִֽם־נִכְלְמ֞וּ מִכֹּ֣ל אֲשֶׁר־עָשׂ֗וּ צוּרַ֣ת הַבַּ֡יִת וּתְכוּנָת֡וֹ וּמוֹצָאָ֡יו וּמוֹבָאָ֣יו וְֽכָל־צֽוּרֹתָ֡ו וְאֵ֣ת כָּל־חֻקֹּתָיו֩ וְכָל־צ֨וּרֹתָ֤יו וְכָל־תּוֹרֹתָיו֙ הוֹדַ֣ע אוֹתָ֔ם וּכְתֹ֖ב לְעֵֽינֵיהֶ֑ם וְיִשְׁמְר֞וּ אֶת־כָּל־צוּרָת֛וֹ וְאֶת־כָּל־חֻקֹּתָ֖יו וְעָשׂ֥וּ אוֹתָֽם		11 And if they are humiliated from all that they have done, the fortification of the house, and its stability, and its exeunts and its entrances, and all its fortifications and all its statutes, and all its fortifications and all its instructions, make known to them, and write for their eyes, that they may keep all its fortification and all its statutes and do them.
ia vam-nclmu mcol awr-ywu xurt hbit utcunto umoxaiv umobaiv vcl-xurotiv vat cl-kuqotiv vcl-xurotiv vcl-torotiv hody aotm uctob lyinihm viwmru at-cl-xurto vat-cl-kuqotiv vywu aotm	59 20	v/am n/clm\v m/cl awr yw\v xvr\t h/bit v/tcvn\tv vmv/xa\iv vmv/ba\iv v/cl xvr\tv v/at cl kq\tiv v/cl xvr\tiv v/cl tvr\tiv hv/dy avt\m v/ctb l/yin\ihm vi/wmr\v at cl xvr\tv v/at cl kq\tiv v/yw\v avt\m

Vav also appears in the middle of roots and words. In this context, it is a weak letter and may appear or disappear or even change to an i. This is outside the scope of prefixes and suffixes.

You will notice that I have rendered every one of the starting letter v (or u) except the one after the atnah as and. In this case, the text looks like a list. But the vav may be rendered as some other conjunction as noted above. Now notice the final vavs sometimes u or o or v. The first two are verbs and the form ending with the u sound is third person plural, so rendered as they. The third is an o sound, the pronoun its - third person singular appended to the singular noun. KJV renders these as the archaic thereof. The same pronoun when applied to a plural noun has the v rather than the o sound. This form occurs 7 times in this verse.

One more example -- in case the music would help us interpret informal inference, or predicates calling for a rendering of then for vav.

Genesis 3:5 -- turns out this is an example where I didn't translate the vav at all.

The music shows how the verse is connected to the prior verse, and the ornament preceding the first vav allows the singer to hear and interpret without any gloss for it. But one could still make an argument for a sound on the vav -- it is its own syllable in the recitative.

כִּ֚י יֹדֵ֣עַ אֱלֹהִ֔ים כִּ֗י בְּיוֹם֙ אֲכָלְכֶ֣ם מִמֶּ֔נּוּ וְנִפְקְח֖וּ עֵֽינֵיכֶ֑ם וִהְיִיתֶם֙ כֵּֽאלֹהִ֔ים יֹדְעֵ֖י ט֥וֹב וָרָֽע		5 ♪C Because God knows that in the day you eat from it, your eyes will be given sight, and you will become as God knowing good and evil.
h ci iody alohim ci biom acolcm mmnu vnpqku yinicm vhiitm calohim iodyi 'tob vry	23 11	ci idy alh\im ci b/ivm acl\cm m/mn\v vn/pqk\v yin\icm vh/i\itm c/alh\im idy\i 'tvb v/ry