EDM is a method for producing holes and slots, or other shapes, by using an electric discharge (spark) to remove unwanted material. It is also called spark erosion. Sometimes it is used to produce a part, such as producing a slot in a very hard metal, and sometimes it is used to "rescue" a part such as removing a broken tap.
The basic idea is to move an electrode very close to the work piece, and repeatedly produce a spark between the two. This is best done while immersed in a dielectric liquid rather than in air, and it helps if the proper distance can be automatically maintained. Note that the electrode gets eaten as well as the workpiece, and some compensation must be made for this. Very good finish can be achieved, though at reduced speed. EDM is not a fast method; some jobs can take days to produce holes, so its use is limited to jobs that cannot easily be done in other ways (e.g. oblong slots or complex shapes, sometimes in very hard material). Note too the work must be conductive so it does not work on materials such as glass or ceramic, or most plastics.
A good overview is in the Metals Handbook published by the American Society for Metals (ASM), volume 3, "Machining" (page 227 of the 8th edition, 1967).
A wire EDM machine uses a wire (usually brass) as the electrode, which passes between guides like a bandsaw blade. The wire may be used only once. As it runs from a spool through the job, it is eroded and reduced in diameter by as much as 1/3. The old immersion system has now been largely replaced by flushing nozzles which surround the upper and lower wire guides. These blast the dielectric through the cut. This system works well, and has the tremendous advantage of being quick to shut off to remove parts, rethread broken wire, etc.
As to the functionality of the wire EDM system, just imagine what you can do with a wire that leaves a .010" kerf through 6" thick stainless steel without heating it up. With the CNC control a taper can be imparted to the workpiece, because even the low-end machines have independently controlled upper and lower guide movement. High-end machines are mostly a way of getting more power (and therefore faster operation); more capacity; and convenience features, like automatic wire threading and a sinker-EDM head to create internal starting holes in hard material.
EDM is well-suited for automated control. In fact, it seems almost the only way to go, which usually puts sophisticated EDM out of the reach of most home shop machinists. Luckily, the fundamentals are simple enough you can get by without computer control at the expense of constantly adjusting the gap manually, as is the case in the Popular Science article.
An easily-built system is described in Popular Science, March 1968, pages 149-153. However, it is electrically dangerous from a shock standpoint, and suggests using kerosene as the dielectric which is a strong fire hazard. This machine was reworked a bit in the January 1991 issue of Home Shop Machinist (HSM), with a few notes in March 1991 (page 11), July 1991 (pages 4-6), May 1992 (pages 10-11), May 1994 (page 44), and May 1995 (page 19). EDM was described in February 1993 in Electronics Now (pages 79-81) along with a list of EDM resources, with a followup in the May 1993 issue (pages 78-80) that references the Popular Science and HSM articles. This machine requires "manual" feed of the electrode. Please note that non-flammable dielectrics should be used, and are available from supply companies such as MSC (I have also heard of using de-ionized water as the dielectric).
A more complex system was described in the July 16 and August 6 1976 issues of Model Engineer magazine (with a letter in the November 5 1976 issue, and a followup in the October 2 1981 issue). In this case a small stepping motor is used to automatically feed the electrode. This is a great improvement, but complicates building the system unless one of your other hobbies is electronics.
From August 1993 to April 1994, Strictly IC magazine published a serial on building another stepper-driven machine. Being newer, it may be easier to build because the parts listed are probably still available. But, again, electronics should be another hobby (or perhaps you have a friend who needs some machining done?). The series also suffers a bit from the author re-designing the system as the series was published, but who ever builds something as published, anyway? (There was also a letter on the project, in February 1995)
In May 1995, HSM started a six-part (one year) series on another complex EDM system, inspired by the Model Engineer series but updated to use contemporary parts. Village Press will re-publish this series as "Build an EDM" which should be available in late 1997.
See also http://www.pe.net/~brettj/edm/ and http://www.edm.com.