A video of this demonstration is available at this link.

The power supply is connected to the coil at left through a tap switch. The coil at right is connected to a galvanometer. When you depress the switch, the galvanometer needle rapidly deflects to one side and immediately swings back to zero. When you release the tap switch, the galvanometer deflects in the opposite direction and again returns to zero. If you reverse the direction of the current in the coil at left (which you can do merely by turning the coil around), this reverses the directions in which the galvanometer needle deflects when you press and release the tap switch. If you increase the distance between the coils, the deflection of the galvanometer needle becomes smaller.

As is demonstration 72.03 -- EMF induced by moving magnet, this demonstration is an illustration of Faraday’s law of induction, which states that a changing magnetic flux induces an electromotive force, or emf, in a loop of wire, or E = -dΦB/dt. (The full-blown law is ∮E · dl = -dΦB/dt.) This emf then pushes a current through the loop. Even though it is the induced emf that is responsible for the current, it is not uncommon for people to speak of the current as being induced by the changing magnetic flux, as a kind of shorthand. For a coil wound so that the turns are close to each other, the change in flux through each turn is the same, and an emf appears across each turn. Thus, for a coil of N turns, the emf across the whole coil is E = -NdΦB/dt. Whereas in demonstration 72.03 a moving magnet provides the change in magnetic flux, in this demonstration, a changing current in one coil provides a changing magnetic flux that induces an emf, and thus a current, in a second coil.

This demonstration is virtually identical to a setup that Michael Faraday used in one of his experiments, about which he wrote, as quoted in Halliday, David and Resnick, Robert. Physics, Part Two (New York: John Wiley and Sons, 1978), p. 771:

When the contact was made, there was a sudden and very slight effect at the galvanometer, and there was also a similar slight effect when the contact with the battery was broken. But whilst the voltaic current was continuing to pass through the one helix, no galvanometrical appearances nor any effect like induction upon the other helix could be perceived, although the active power of the battery was proved to be very great. . . .

A (steady) current in one coil produces a magnetic field around the coil, whose magnitude at the center of the coil is B = (μ0/2)(Ni/r), where N is the number of turns in the coil, i is the current and r is the radius of the coil, and whose direction is given by the right-hand rule. If you curl the fingers of your right hand in the direction of the current in the coil, your thumb points in the direction of the magnetic field (north). (See demonstration 68.13 -- Right-hand rule model.) By convention, current refers to the flow of positive charge, which is in the opposite direction to electron flow. At points far from the coil (x r), B = μ0NiA/2πx3,where A is the area of the loop and x is the distance of the point along the axis perpendicular to the loop.

The magnetic field at the second coil depends on the distance between the two coils, and their orientations with respect to each other. If they are parallel, then the field is perpendicular to the plane of the coil, and its magnitude depends on the distance between the coils. B is greatest when the coils are close together, and falls rapidly as the distance between them increases. The magnetic flux through the coil is Φ = ∫B · dS, where the integral is over the area bounded by the coil. Since B is perpendicular to the coil, and the coil is circular, the flux through it is Φ = BA, where A is the area of the coil. The unit of magnetic field is the tesla, which equals 1 N/A·m. 1 tesla also equals 1 weber/meter2. The weber, which also equals 1 V·s (= 1 N·m/A, or 1 kg·m2/A·s2), is the unit of magnetic flux. In this demonstration, it is the change in magnetic flux, dΦB/dt, that is important. As long as the current is constant (zero or maximum) in the first coil, there is no change in magnetic flux through the second coil. Thus, there is no induced emf in the second coil, and thus no current, and no galvanometer needle deflection.

When you depress the tap switch, the current in the first coil goes from zero to some steady value (~5 A). With the coils as shown in the photograph, the current in the first coil flows in the counterclockwise direction as you face the coil from the left. This produces a magnetic field for which north points to the left. When the current makes the transition from zero to 5 A, the change in magnetic flux through the second coil induces an emf that causes a current to flow in the direction that produces a magnetic field for which north points toward the right. This is a clockwise current as you face the coil from the left. With the coils as shown, this produces a current that flows into the positive terminal of the galvanometer, and thus causes a deflection of the galvanometer needle in the positive direction (to the right).

Once the current begins to flow in the first coil, it becomes steady, and so does the magnetic field it produces. The magnetic flux through the second coil is thus constant, so when the galvanometer needle deflects to the right, it immediately returns to zero.

When you release the tap switch, the current goes to zero, and so does the magnetic field. The resulting change in magnetic flux has north pointing to the right. This induces an emf in the second coil, in the direction that produces a magnetic field for which north points to the left. This is a counterclockwise current as you face the coil from the left. This current enters the galvanometer through the negative terminal, and thus causes a negative deflection of the galvanometer needle (to the left).

Again, once the current begins to flow in the first coil, it becomes steady, and so does the magnetic field it produces. The magnetic flux through the second coil is thus constant, so when the galavanometer needle deflects to the left, it immediately returns to zero.

You can show that when you increase the distance between the coils, the deflection of the galvanometer needle becomes smaller. When you reverse the direction of the current in the first coil, this reverses the direction of the change in magnetic flux that occurs when you press, and when you release, the tap switch. In turn, this reverses the direction of the emf induced in the second coil, and thus the current and the direction in which the galvanometer needle deflects, when you press, and when you release, the tap switch. You can show this merely by flipping the orientation of the first coil.

If you wish, you can also hold the tap switch down and demonstrate the change in magnetic flux that occurs through the second coil when you move one coil away from or toward the other.

The linking of two coils by the magnetic flux produced by a current in one of the coils is called mutual induction (hence the title of this demonstration; this is different from self induction, in which the magnetic flux linkages occur within one single coil). If we call the coil connected to the power supply coil 1, we can define the mutual inductance of coil 2 (connected to the galvanometer) as M21 = N2Φ21/i1, or M21i1 = N2Φ21. If i varies with time, then M21di1/dt = N2dΦ21/dt. The right side of this is the emf in Faraday’s law, above, without the minus sign, so we have E2 = -M21di1/dt. If we now connect coil 2 to the power supply and tap switch, and coil 1 to the galvanometer, we find E1 = -M12di2/dt. It turns out that M21 = M21, and we can drop the subscripts and call the mutual inductance M. The SI unit for M is the henry, which equals 1 V·s/A, or 1 weber/A. The value of M, and how one calculates it, depend on the geometry of the coils and their orientation with respect to each other.

References:

1) Halliday, David and Resnick, Robert. Physics, Part Two, Third Edition (New York: John Wiley and Sons, 1978), pp.720, 770-4, 805-6, 886.
2) Sears, Francis Weston and Zemansky, Mark W. College Physics (Reading, Massachussetts: Addison-Wesley Publishing Company, Inc., 1960), p. 612.