How do I choose a bicycle fork? I'm putting together a bicycle on a budget from a frame I was given and various parts from a bike with a broken frame. The only part of any expense that I really need is the fork, and I want to get this via an auction (likely Yahoo auction Japan). 

The frame itself is a women's Scott Tacana. Something like this:



I'm not necessarily looking for shocks, I really just want something cheap that fits. Are there different diameters, tube lengths etc, that I need to be aware of?
There are only a few measurements you need to be aware of when purchasing a new fork, particularly if you're avoiding suspension.  If you don't need suspension, don't get it - it will only add cost and complexity, and a cheap suspension fork will be much worse than a rigid fork.

The basic measurements you need to be aware of are:


Headset type.  Not truly a measurement, but there are two types of headsets - threaded and threadless.  The photo you show is of a bike with a threadless headset; the fork's steerer tube will be a smooth, unthreaded tube and the headset is tensioned with a bolt from the top and the clamping action of the stem.
Steerer tube diameter. Also two main options, 1" (25.4mm) and 1 1/8" (28.6mm, also known as 1.125" or 9/8").  Nearly all threadless headsets use 1 1/8" steerers. You can verify that the frame you have is for that size by measuring the inside of the head tube, it should be 34mm.  If you already have a headset installed, you can verify that the inner diameter of that is 28.6mm.
Steerer tube length. If you are purchasing a new (or "uncut") fork this is extremely unlikely to be an issue, but on a used fork you'll need to check. The steerer tube needs to be long enough to extend above the top of the headset with room for the stem to clamp fully.  Too short and it can't be used with that frame.  Too long isn't an issue, since you can easily use spacers and/or cut it to length.  If you measure your stem clamp, your headset, and the length of your head tube and add a few mm for a safe margin you've got a bare minimum steerer length you'll need.
Wheel size and brake type. The bike in the picture has 26" (559) wheels and is designed for cantilever or v-brakes. The position of those brake mounts is specific to the wheel diameter, so you'll want to make sure it's for the right size wheels.  If you switch to disc brakes, you can omit this concern, but not entirely, because of...
Axle-to-crown length. This is the distance from the center of the axle to the top of the fork crown, just below where the steerer tube starts. The frame's geometry was designed around a certain distance here, and changing it dramatically will alter the handling by increasing or decreasing the angles relative to the ground.  For mountain bike forks this information is more difficult to come by than for road bike forks, but if your bicycle originally had a suspension fork the rule of thumb for rigid is to find a fork that is "suspension corrected" for the amount of travel your bike originally sold with.  In the case of the Scott Tocana it looks like most models shipped with an 80mm suspension fork, so an 80mm suspension-corrected rigid fork should keep things normal.


Basically, given the bike you've shown, if you look for an '26-inch fork, 80mm suspension corrected, 1 1/8" uncut steerer, v-brake posts' you will probably find something suitable.