There are a handful of ways available to the hobbyist to turn your own designs into PCBs. They yield results of different qualities, where the quality seems to be inversely proportional to the amount of mess you make (in most cases), and amount of money you spend (in all cases). I'll talk a bit about each, and then compare them all at the bottom of the page.
Any process that involves making your own board will have a number of steps in common. At a high level, here's what you're doing:
1. Procure a bare board (coated with a thin layer of copper on either one or both sides). Most methods will use a plain board; photolithography requires one coated with special light-sensitive chemicals.
If you have a plain board, scrape off any burrs along the board edge (you want a flat copper surface; I use a fine file for this), and clean it well to remove oxidation and finger oils. I start with fine steel wool, follow up with denatured alcohol to remove any oils or grease, and finish by buffing with a very clean towel. From this point on, you'll want to handle your board only by the edges to avoid getting finger oils on it.
2. Design your circuit. Depending on how you plan on actually producing the board (read on...), your design will take one of a number of different forms -- a hand-drawn set of lines on paper, a computer-drawn diagram, a design file you'll send off to a manufacturing house...
3. If you'll be producing the board yourself, transfer your design of desired copper traces to the plated side(s) of your board; the transferred traces are resistant to your etching liquid (more on this later). Most home-brew board production methods differ only in how they accomplish this step. If you are generating a design via computer, you'll have to put some thought into which way your printed design faces (i.e., printed "right way up" vs. mirrored). There are enough ways to approach this that I split this information out onto its own page here.
4. Etch the board you've traced -- here, an etchant chemical removes all non-masked copper; after it's done, give the board a good wash under running water to remove all traces of the etchant. In most cases, the etchant will either be Ferric Chloride or Ammonium Persulfate (Ferric Chloride is more popular). These are available in both liquid (i.e., premixed) and powder form; the powder is generally quite a bit cheaper, but requires care when mixing.
Also note that etching proceeds faster with (1) warmer etchant, and (2) agitation. Along with saving you time, fast etching also produces better edge quality and consistent line widths, so fast is good in this step. I pre-heat Ferric Chloride etchant in the microwave for 40 seconds or so (you want it hot enough to be barely-comfortable to the touch), and slosh it around by hand as it's doing its work. An old plastic freezer container (with lid) is good for this (it allows for vigorous agitation, without making you wear any of the etchant). You can keep the etchant warm by putting the etching tray inside a larger tray or sink filled with boiling water.
Note that you don't want to get Ferric Chloride solution too hot, since it will start generating Hydrochloric acid fumes if you do (very corrosive, bad for eyes and lungs).
Use lots of running water when you dispose of your used chemicals (and when you rinse off your finished board), as etching chemicals will stain plumbing and fixtures (and clothes, and exposed skin...). Ferric Chloride, in particular, attacks most metals -- including stainless steel. This stuff will also cause permanent eye damage, so be careful out there...
I've heard good things about Sodium Persulfate (it's a clear solution so you can see what's going on, is non-staining, doesn't generate any hazardous fumes), but haven't tried it myself.
5. Cut the board to final size and shape, and drill holes in the board for component leads. These need to be very small holes (about 0.8 mm); occasionally you can find resharpened carbide industrial bits for sale from surplus houses -- this is a good way to save money as resharpened bits cost about $1 US each, whereas new ones cost about $10 US.
You'll find these bits described in a variety of fashions -- fractions of an inch, decimal inches, decimal mm, even "numbered" sizes -- I've got a conversion table on a separate page. Ideally, you'll want to buy carbide bits (they're stiffer, and thus "wander" much less than steel bits), and you'll want to buy a number of them (they break pretty easily, particularly if you don't have a drill press, and so are drilling by hand).
6. Carefully scrub off the mask (with fine steel wool under running water), and populate the board (i.e., solder on your components). You should only scrub off the mask when you're ready to start soldering, as the copper traces oxidize quickly (i.e., within a few days).
After the board is populated (i.e., all the components have been soldered on), I usually follow up with a quick coat of spray polyurethane varnish -- this keeps the shiny copper traces looking shiny, and (more importantly) provides a bit of insulation against "shorts" due to stray wires brushing up against the board.
