The term "hey" goes back to at least the 16th century. The earliest reference that I know is in Orchesographie (1589). "Branle de la Montarde" (a version of which is now danced as "The Horse's Branle") has "le premier fait une haye" (the first person dances a hey). In this particular hey, the first person just weaves to the end of the line, passing in front of the ladies and behind the men; no-one else moves.
The next dance in the book is "Branle de la Haye" in which A, B and C dance a three-person hey as we know it today.
Haye, or haie, meant a hedge. Imagine a hedge made by weaving branches horizontally through posts and you can see the weaving concept. It was also used to mean a hedge made of soldiers (e.g. lines of soldiers greeting visiting dignitaries), again, giving the opportunity to weave between them.
I understand that there are references to the hey as a weaving figure going back even further.
William Hogarth: "The Analysis of Beauty" (1753):
"The lines which a number of people together form in country or figure dancing, make a delightful play upon the eye, especially when the whole figure is to be seen at one view, as at the playhouse from the gallery; the beauty of this kind of mystic dancing, as the poets term it, depends upon moving in a composed variety of lines, chiefly serpentine, govern'd by the principles of intricacy, &c. the dances of barbarians are always represented without these movements, being only composed of wild skipping, jumping, and turning round, or running backward and forward, with convulsive shrugs, and distorted gestures.
"One of the most pleasing movements in country dancing, and which answer to all the principles of varying at once, is what they call the hay; the figure of it altogether, is a cypher of S's, or a number of serpentine lines interlacing, or intervolving each other..."
Reels
The Scottish created dances based solely on the hey. They called them reels. The scotch reel consisted of alternate heying and stepping (fancy steps danced in place) by a line of three or four dancers. Susan de Guardiola's page on what Jane Austen might have danced gives some more details. The dance is still popular today in forms such as the Dorset Four-Hand Reel. See also Three-Hand Reel and Five-Hand Reel.
These days we use the terms "hey" and "reel" to mean the same thing; which word you tend to use usually depends on which genre of dance you learnt first.
Rules for Heys
Many dancers worry about which shoulder or hand to use. Here are some concepts that might help:
Rule #1: Don't crash!
If you are starting a hey then try to use the shoulder that the caller taught you to use, but if the other dance is obviously going to go the other way, then go with the flow. The dance will still work in most cases.
If you are not initiating the hey then you should still move; you can probably see someone's back, and, past them, someone's face - move sideways slightly, in the same direction as that person.
If you get to the end of a line then keep your eyes on the last person you passed to that you turn the easy way to re-enter the hey.
If you get to the end of the line then, while you are turning, you will have missed one of the passings. That means you must use the same shoulder/hand twice. If in doubt trust the person coming out of the hey.
At the end of the line, don't turn on the spot - that would put you too close to the person in front of you. Make a small loop to use up the time.
It is a weaving figure; never get too close to the person in front - someone wants to come between you.
Hey Variations
The chief characteristic of a hey is that it is a weaving figure. But how many people are involved and what formation they are in varies tremendously. Here are over 25 different forms of hey.
La Haye
One person weaves along a line of stationary dancers.
Straight Hey
A number of dancers in a straight line simultaneously weave the line. This is the most common hey or reel. This can be done with hands or without.
Grand Hey
Two intersecting Straight Heys. Where the lines cross is a junction; the dancers have to follow the path of a very small circle around the junction in order to avoid crashing.
Circular Hey
An even number of dancers in a circle simultaneously weave the line. Normally done with hands, as in a Grand Chain (Grand Right & Left). Without hands it is known as Weave the Ring.
Progressive Hey
A Hey which is started by one couple, others join in as the leading couple reach them.
Figure Eight
Two people weave around two stationary people, starting diagonally between the stationary people.
Double Figure Eight
A Figure Eight where the other two people join in by casting then doing the same figure simultaneously. We think this is the figure in Punch Alive (1726) where it is called "Hey all four".
Grimstock Hey
Two Straight Heys. Tops face down, everyone else faces up; Tops start down the middle. The Heys mirror each other.
Adderbury Hey
Two Straight Heys. Tops face down, everyone else faces up; Tops start by passing right shoulder with the Middles. The Heys are parallel to each other.
"Morris Hey"
Two Straight Heys. Tops and Bottoms turn out, Middles come up the centre. This is a country dance term for this type of Hey. There are of course many Heys in the Morris, many of them listed here.
Slip Hey
Any of the above, done while facing across the set, and dancing sideways throughout the Hey.
Lichfield Hey
Two lines of four dancers. The rule is: if you are heading towards the centre keep facing the centre, if you reach the end then face across. In fours, First Corners pass right shoulder, Second Corners pass right shoulder, pause, everyone pass left shoulder. Repeat three more times to get home. You are following the path of a toffee paper or Christmas cracker.
Shetland Reel
A three person Straight Hey done by six dancers. Each couple acts as a single unit with the leader staying in front throughout the hey. Lots more information here.
Dolphin Hey
A three person Straight Hey done by four dancers. The middle couple act as a single unit, but the leader goes wide at the end of the line, allowing the trailer to turn in front of them and take the lead.
Variant: The couple acting as a couple take a Poussette Hold and stay that way throughout the Hey, taking it in turns to be going backwards.
Couple Hey
A three person Straight Hey, done by six dancers, each pair coupled with their arms around each other’s waists.
Serpentine Hey
Four couples in two straight lines. Top couple and third couple face down, second couple and bottom couple face up. The person on the left of each couple is a leader and holds nearest hands with their partner. Basically the leaders dance a hey dragging their partner behind them. In fours: circle CW half way, middle four back circle (without joining hands with the other couple) AC half way. Repeat as many times as you wish making the flow nicely serpentine.
Sheepskin Hey
Three dancers weave the line around three stationary dancers. Every time the trailing person pass the middle stationary person they go around the middle person and take the lead.
Zig-Zag Hey
This is a single Straight Hey done in two lines. Pass the person diagonally on the right by the right shoulder; pass the person opposite by the left shoulder. Repeat as necessary.
Figure Eight Hey
Four couples in two lines. In fours: pass the person beside you in the line by the right shoulder; ends pass left shoulder while the four middles Star Left (without hands) half way. Repeat three more times to get home.
Spinoff Hey
Progressive Straight Hey for four dancers. Whenever you get to the middle you reach out with the nearest arm to the other dancer’s waist and you both spin past each other. The track remains unchanged.
Triangular Hey
Three dancers in a triangle. #1 does three changes of rights & lefts: with #2, #3, #2. Then #2 changes with #3. #2 and #3 wait when not involved in a change. Everyone is back where they started.
Ricochet Hey
Progressive Straight Hey for four dancers, the middles start facing each other and start the hey by the right shoulder. Specified dancers, instead of passing each other in the middle, meet with both palms and push back to the side they came from, returning backwards along a different diagonal, then moving across (think pizza slice).
Tapsalteerie Hey
First couple pass, second couple ricochet, first couple ricochet, second couple pass.
Gypsy Hey
The dancers are in a circle or line facing their partners. All dancers dance a three person hey with the person in front and the person behind: Gypsy your partner right shoulder, gypsy your corner left shoulder.
Variant: Extended Gypsy Hey: Pass your partner right shoulder and gypsy right shoulder with the next, pass your partner left shoulder and, gypsy left shoulder with the next.
Celtic Hey
Line of four, men on the outside facing their ladies. Start a Right Shoulder Hey - but Men Gypsy Left 3/4 WHILE Ladies Orbit C 1/4 to make a new line of four at right angles to the first line (facing partner); Start a Right Shoulder Hey - but Ladies Gypsy Left 3/4 WHILE Men Orbit C 1/4. This is equivalent to two Square Dance Spin the Tops, but without hands.
Bombast
Four dancers in a square: First Corners cross the set and spin right while Second Corners Gypsy Right 1/2, moving one place AC around the square, and spin right. Repeat three more times to get home. In the initial movement you are passing someone by the left, then middles pass right, so it can be considered to be a hey.
Wizard’s Walk
This is a Straight Hey in a Longways Set, the dancers going forwards two places then backwards two places, without turning. The original dance has the #1s starting up the outside, with the #2s oscillating out of their way. For the Lizard Walk variant the #2s can move up the set and back to share the work with the #1s.
Schiehallion Reel
A cross between a Grand Hey and a Clover Leaf. See here.
Lock Chain Swing
Grand Chain, but instead of pulling past you do one and a half forearm turns.
Grand Leer
A Square Dance move where one gender dances all the way around the square weaving between the other dancers of the other gender. This can also be done as a Circle Mixer move: one gender weaves the ring, then the other gender weaves in the same direction to meet them again.
For an "in-depth study of the Hey as described in “The Dancing Master” of John Playford, Henry Playford and John Young, and interpreted by Cecil Sharp and others", please see Colin Hume's The Hey