A transformers is basically a metal core with wire coils wrapped around it. When a signal is applied to one winding, it develops a signal on the other winding that steps up or down the voltage by the ratio of the turns. If the ratio is an increase, there is a voltage gain. The big advantage of using a transformer for voltage gain is that they are quiet -- they do not exhibit the hiss and other noise of a transistor or tube circuit. Additionally, because they are passive devices, they do not require a power supply of their own.
However, a transformer cannot just be thrown into any circuit. Transformers generally work best when driven by a low impedance. Additionally, while they provide gain by the winding ratio, they also increase impedance by the square of the ratio. In other words, if one drives the primary of a transformer with a 1 to 2 (1:2) winding ratio with a 1V signal and a source impedance of 100 Ohms, the output from the transformer's secondary will be 2V, but the source impedance will be 400 Ohms. This is increasingly important as the impedance and the turns ratio increase. For instance, driving a 1:4 transformer with a 1000 Ohm source results in an output impedance of over 16,000 Ohms.
Finally, small signal transformers, the sort that one might use for signal voltage gain, generally to not tolerate DC offset. It depends upon the transformer exactly how rigid that requirement is. For a large steel cored transformer, a few millivolts of DC (say .003V) might prove incidental. But, for a mu metal transformer even that can lead to saturation which will, at the least, limit the amount of signal the transformer can pass, and at worst lead to terrible distortion. Amorphous cobalt cores, such as those used by Lundahl, are even more sensitive, and for those the rule really is 0 DC.
This brings us to the Black Diamond. The idea here was to use a buffer to drive a stepup transformer followed by an output transformer. The input buffer provides a high input impedance for the amplifier. It also provides a low source impedance for the transformer. The output buffer drives the headphones. And the transformer, of course, provides the gain.
While this seems simple enough, getting the buffer right is critical. The simplest buffer is an emitter or source follower.
With an emitter or source follower, input impedance is reasonably high, output impedance is low, and distortion is pretty good, particularly when loaded with a constant current source (CCS). However, power supply ripple rejection (PSRR) is terrible requiring a very good power supply. Additionally, with both BJTs and FETs, one will need output capacitors without fairly heroic measures, and with BJTs input caps are necessary, too.
Theoretically, one could add a PNP emitter follower immediately after the NPN follower, and hope that the Vbe's of the two transistors cancel out. With a bipolar supply and clever biasing, this could at least minimize the offset. The usual way of doing this also involves adding a PNP to NPN pair in a circuit commonly referred to as a Diamond Buffer.
Not only does the addition of the second pair help cancel even harmonic distortion and increase PSRR, but it also, in the case of BJTs, minimizes offset. Well, it is supposed to.
Unfortunately, in practice, none of this works as well as one might like. Output offset is still non-zero, and no amount of adjusting will fix it as it will always drift. However, like a simple emitter follower, the circuit can be improved by loading the input transistors with CCSes. And by adding the CCSes, not only is performance of the circuit better, but it allows the output offset (the more egregious offset problem) to be addressed by use of a servo. Further, in theory anyway, the input offset should be nulled by the diamond buffer's structure.
As drawn, this actually won't quite work as the DC resistance of the biasing LED must be increased a bit to allow the servo to do its thing, whether through some series resistance, or by replacing it with a resistor that is bypassed by a capacitor.
A SPICE simulation reflects the quality of the circuit, and the effectiveness of the servo.
As for the input offset, while SPICE would have you believe it will be zero, in practice, no amount of transistor matching  will actually make it so. On the amplifier input side, it is low enough that with a fairly low impedance volume pot, there is not much of an issue. There is a slight scratchy noise when turning the pot from 0 to on, but it is otherwise quiet. Whether this is a concern is probably a matter personal tolerance. If one is a DIYer and used to putting up with glitches, it is not a big deal. On a commercial amp, it is probably a deal breaker.
The real issue seems to be on the input side of the output buffer. There the offset drags some small amount of current across the output transformer. With a steel, or even nickel core transformer, the current does not seem to be detrimental as it is very very small. However, I used (see part 2 of this article) Lundahl LL1544a stepup transformers which have an amorphous cobalt care, and do not tolerate any offset. For these, it seems to be enough to create some harshness in the sound.
The fix for this will be in Part 2 along with an actually assembled amplifier ...
 This article will not get into the details of how a transformer works, or which one is appropriate for any particular use. For that sort of background detail, read Audio Transformers by Bill Whitlock Chapter 11 From the "Handbook for Sound Engineers" Third Edition. (3MB PDF)
 I have not yet tried factory matched transistors like the THAT corp.'s 340P14-U.