The PCB program used to lay out the board is called simply PCB, and is available from the PCB Sourceforge site. If you run Linux, chances are that PCB is in your distro's repository - Fedora users can find it by just clicking on Applications -> Add/Remove Software, then searching for 'pcb', which will download and install the program.
If you can't run PCB, I have provided PostScript files that can be sent directly to a printer (for either the toner transfer method of making PCBs, or the photographic method). Gerber files are provided so that the boards can be made at a commercial PCB factory.
The board layout is double sided, but optimised for home construction. This means I'm assuming you don't have through plated holes, and will be soldering by hand. All the ICs therefore are soldered on the bottom side of the board - IC pins do not double as vias on this board. This makes it easy to use chip sockets for all ICs. It does mean there are a few extra vias, though. The only components that need soldering on the component side are the LED anodes (which are easy to solder - just have the LEDs stand off the board about quarter of an inch), one pin of the flash chip's decoupling capacitor, and the SOT-23 MOSFETs.
If you've not yet made a double sided board, don't be too intimidated by it - using the toner transfer method with cheap Tesco's Value glossy photo paper, it was easy to get the registration right. The signal tracks on this board are 10 mils wide with a minimum spacing of 10 mils - in practise, I've found this works out just fine with the toner transfer method and cheap photo paper. I have never tried the UV photographic method so I can't comment on that - but I presume the resolution is better for the photo method.
"True video" PostScript files are provided for the toner transfer method and some photo process methods, "inverse video" files are provided for some photo process methods (the type you'll need depends on the photo resist you're using). For the toner transfer method, you'll need the non-mirrored version for the solder side, and the mirror image for the component side. It's worth printing off a copy of the layout page to help with assembly (so you know which holes are vias, chip pins, jumper pins and LED pins so you know which drills to use when drilling.
Gerber files are provided for having the board made at a PCB factory. The zip file includes the CNC drill file, solder masks, paste masks (I doubt you'll need those though) and silkscreens as well as the electrical layout. I've never had a PCB built for me by a factory - if you do, please let me know how it turns out (a photo would be nice too). If the PCB fabricator has preferred drill sizes, you can let me know what they are and I'll make Gerber files with those drill mappings. Some PCB makers might not like the copper fill areas - I can also prepare files with those removed.
The components page mentions this, but I'll mention it again: the LEDs required for this board must have a built-in resistor. These are sold as '5 volt LEDs with a built in current limiting resistor' at most electronic supply houses. Using LEDs without the resistor (without adding extra discrete resistors) will cause bad things to happen.
Don't be intimidated by surface mount - this was the first time for me, too. I found what worked for me was putting a tiny drop of Superglue on the PCB where the transistor would land, then carefully placing the transistor with tweezers, and letting the glue set. Then solder with 0.5mm solder (which I use for all electronics soldering), and a 1mm tip. There are some tutorials on how to do surface mount by hand at Spark Fun Electronics - they use a different method to me (tacking a pin down first with solder).