I was inspired to build this after seeing Carl Willis' wonderful video of his own detector - http://youtu.be/-8GlzUjYazs. The principle is relatively simple - a high voltage is applied between a grid of fine wires and a plate, and any ionizing radiation, such as α-particles, is sufficient to cause breakdown of the air and small spark is formed. By laying out the wires over a reasonable area, it's possible to even obtain a 2D picture of the radiation distribution. Although it doesn't look nearly as nice as Carl's one, this can be lashed up in under ½ hour and works well enough to demonstrate the effect clearly.
The baseplate is a piece of 6mm acrylic. The cathode plate is 1.2mm aluminium and is superglued to the baseplate. The anode wires are 0.1mm diameter copper, obtained from a piece of extra-flexy test wire. Finer wires work much better; I had previously tried 0.2mm steel wires, and absolutely nothing happened. The anode wires are supported on two pieces of thick PCB material which are also superglued to the baseplate. The PCB material is 2.3mm thick, but it can also be normal-thickness PCB built up. Note the fine saw cuts made into the PCB material between the wires - these thermally isolate each wire from the adjacent ones, allowing each to be soldered without the others melting off. Wire spacing is 2mm. Tin the PCB first, solder one end of the wire, then stretch it lightly over and solder the other end. The small amount of thermal contraction produced pulls the wire nice and tight. Spacing between the wires and the cathode is about 1mm.
To operate it, I used 5kV (the limit of my HV supply) and I connected the detector directly to the output. Carl said that he used a series resistor, but I think the internal impedance of my supply is high enough that I don't need one. Start with one anyway (20MΩ or so) to see if it's needed. The voltage should be turned up as far as possible without forming sparks. Then, when an α-particle source is brought within about 4cm of the detector, sparks start occurring between the wires and the plate, indicating the presence of radiation. It's a really nice effect.
The source I'm using is Americium 241 from a smoke detector - see here for how I made the little holder.
Here's a video showing the operation and the effect of a piece of paper on the α particles - it blocks them completely.