There are several instructibles on this general topic already, but this is just the way I did it for this style of guitar. Doing a hollowbody or Les Paul-style guitar takes different steps and techniques, so this won't work fully for any style of guitar other than a flat-top, solidbody, bolt-on neck guitar. The dimensions given are for this body only; if you want to make a Strat or Tele or whatever, you may need more or less wood accordingly. The shape of this one is a hybrid of a Fender Jazzmaster and Stratocaster.

Materials!

Wood: 2 pieces, 8/4 (~1.75" finished) lumber; 8 x 20"
Any hardwood can work; maple, mahogany, alder, poplar, or basswood are traditional. The two things to keep in mind here are weight and finishing technique. You could make a body out of oak if you wanted; but it would weigh a hundred pounds. I don't fully buy into the wood having a drastic effect on the tone of an electric guitar, beyond harder woods--maple, ash--giving a bit brighter sound and softer woods--mahogany--giving a bit more mellow sound. This site is a good resource for info on the tone "characteristics," finishability, and weight of different typically used guitar woods.
Wood glue
Titebond will work fine for this, don't need anything fancy. Gorilla Glue will give the strongest bond, but it's also a pain to work with because it expands from the joint dramatically while drying and requires extensive cleanup; it's also a bear to sand. Do not recommend.
Grain filler (optional)
If you use something with an exaggerated and open grain--mahogany, ash--you'll want to fill the grain before finishing, or you'll be dumping gallons of finish on to get it smooth.
Paint/dye (optional)
I used latex paint here; this is not a traditional choice, but it's the easiest way to get a custom color, as any home improvement store can mix it up for you. If you go that route, a sample size container is plenty; if you use spray paint of some sort, 1 can should do it for a more dense wood like maple, 2 for something more porous like mahogany.
Clear coat: 2-3 cans
I'm using gloss lacquer; polyurethane is easier to apply (requires fewer coats) and more durable, but it goes on thick, and I don't like that look. Most modern guitars have poly finishes, though.

Tools!

Jig saw or band saw
Router
Drill press
Drill
Belt sander
Orbital/finishing sander
Forstner and standard drill bits
1/4, 1/2, and 1/2 roundover router bits
Clamps
60, 120, 220 sandpaper
0000 steel wool

Step 1: Prep the Wood

Cut your lumber to length (20") and get a good smooth, straight edge on 1 long side of either piece. If you get a body blank or a piece of lumber wide enough to not have to do this, skip to the next step.

Run a largish bead of glue down one of the prepped long sides, then press the boards together.

Clamp them tight, but not too tight. A little glue run-out is good, but don't press it all out.

Follow the instructions for whatever glue you use; for most wood glues, leave the pieces clamped for an hour, then let the combined piece sit for minimum couple of hours more before doing any work on it.

Plane/sand both faces smooth.

Draw your body outline on, including where your cavities/neck pocket/holes will go. I used a plan for a Fender Jazzmaster as my starting point, then tweaked to fit my vision for it. The PDF plan I used is attached; plans for pretty much any other body style can be found online. Or just make one up!

A standard neck pocket is 5/8 x 2 3/16 x 3"; using this will allow you to use an off-the-shelf replacement neck if you want. You can make it a different size if you're building your own, obviously. The cavity for the Strat-style jack plate is approx. 1 x 2 1/2", and needs to be minimum 1" deep. The pickup/control cavity size depends on what kind of pickguard or control plate cover you're using, as well as what kind of pickups. Humbuckers need a deeper cavity than single coils, and P-90s or Jazzmaster-style pickups actually mount to the bottom of the cavity, not the pickguard or pickup ring, so need to be pretty close to the pickup specs. I highly recommend--experience taught me this--getting all your bits and bobs before you start so you don't find out later you've screwed something up.

Fender62Jazzmaster_RevA.pdf
Download

Step 2: Routing and Cut Out

You can route the entire thing after you cut it out if you want, but at least route the neck pocket before cutting. Once the body's cut out there's not much wood up there for the router to sit on, and you won't have a straight edge to use for a router guide. It's vital the neck pocket edges be as close to perfect as you can get them.

The routing can be made significantly easier by drilling out the bulk of the wood with a drill press and forstner bit. I routed out more wood than is strictly necessary in order to bring the weight of the final product down. (Maple is extremely heavy, probably one of the heaviest woods normally used for guitars.)

Before drilling the various bridge screw holes, check your layout. Measure your neck, nut to butt, then subtract that distance from the scale length; the difference is how far back from the bridge-facing edge of the neck pocket the break point should be. (Break point being where the strings actually hit the bridge saddles.) If using a standard Fender replacement neck, the scale length (distance from nut to bridge) should be 25.5"; you can verify it by measuring nut to 12th fret, then doubling that (for example, a standard Strat neck will be 12.75" from nut to 12th).

After you've routed, go ahead and drill the through-body holes and ground wire hole. On the ground wire hole, put a couple layers of tape down so the drill's chuck doesn't chew up the wood if you slip. For the string holes, don't do this with a standard drill unless you've got a level on the back. I did, and it created an issue (as you'll see in later pictures) that had to be corrected. If you have a drill press, make a jig:

Take a piece of your body cut-off and attach the bridge to it with the mounting holes/screws
Drill the string holes all the way through the block with the drill press
With longer screws of the same size, mount the jig block to the body
Drill the string holes with a hand drill; this way the bit is deep enough in to not wander once it actually hits the body wood (be careful not to start drilling a new channel while moving the bit through the jig; easiest way to avoid this is to run the drill in reverse while inserting it)

Cut out the body with a band or jig saw. Be careful not to cut inside the line.

Step 3: Clean It Up, Carve Accents

Clean up the edges so they're nice and smooth. I use a combination of the round end of my belt sander for roughing down open areas, and drum sander drill attachments (2" and 1/2") with my drill press for tighter curves. This can also be done (more quickly) with a router, preferably on a router table. Be careful around the neck pocket.

Carve the forearm, belly, and right bout (the area to the right of the neck at the top of the body, where your hand sits while playing on the high frets) divots:

Mark the outline of the "cuts"
Using a 1/4" router bit, carve a series of "steps" to rough out the shape and depth of each area
Use the belt sander to grind the slopes smooth on the forearm and belly cuts; for the right bout, I'd use a dremel on it to rough it, then finish with a rasp—it's too easy to get carried away or for the belt sander to get too eager and ruin your neck pocket

If you want to make the neck join a little more comfortable, use the same process to add a slope to the body portion of the joint. Again, be very careful here, it's very easy to ruin the whole project with a slip-up here. Make sure you leave at least 3/4" thickness.

Drill the strap post and neck mounting holes, and make sure everything's good for fit. Neck screws are typically #8, 1 3/4"; I'm using #10 on this one, because that's what I grabbed by accident, and 1 1/2" because I'm countersinking the screws.

Sand everything to 220 or higher.

Step 4: Painting/Finishing

Clean all the sawdust off the body, then apply the color coat. I brushed on; it was very cold, though, so the paint was thicker than it would be normally, and brush marks were an issue. Found after the fact that this can be alleviated by heating up some water, then sitting the paint container down in that before use. This trick will work for spray colors/clear coats, too.

Sand the color coat smooth with high-grit sandpaper (400). Touch up as needed.

Clear coat it; I used gloss Minwax lacquer. Polyurethane is easier to apply and is more durable, but I wanted to "relic" it, and lacquer's much easier to do that with. Thin coats, make sure to over-spray (move the can past the edges) to minimize runs. After the first couple coats, start lightly dulling the finish with the steel wool between coats. This will help create a nice glassy look. Be sure you get all the dust and steel wool bits off before spraying again.

After the clear coats, if you're not going to relic the body, let it sit for a week or so and cure. Then polish 'er up.

If you are going to relic it:

Sand back the finish in spots where there would be wear naturally (the arm "rest", around the jack, the middle of the back, etc.); make sure it doesn't look planned out. For a more general wear, slide it around a little on a relatively smooth concrete surface, like your garage floor. Make some dings by "slipping" with a screw driver like you would while adjusting the bridge or swapping a pickguard. And some grunge by putting some brown or cordovan show polish on a paper towel, wiping it on, immediately wiping it off, then clean it up by spraying Windex onto another paper towel. Don't overdo it.

Step 5: Hardware Install/Wiring

Install the string ferrules on the back, the bridge, and the strap pegs. Before installing the bridge, make sure to run your ground wire through to the control cavity; the wire doesn't need to be actually attached to the bridge, but make sure to strip enough of it to get good contact and stay in place when the bridge is screwed down.

Install your pickups and electronic bits into the pickguard, and wire it up. Standard Strat wiring diagrams are all over the place online, but I've included the one I used. If you're new to soldering, this instructable, while not specifically about guitar wiring, is very, um, instructive.

This Instructable was sponsored in part by The DIY Outlet, who provided a 2-in-1 Hot Air & Iron Soldering Station that was invaluable in this project. A good multi-temp soldering iron can cost around $40, and a multi-temp heat gun that much or more; this provides both in a compact package. I used the heat gun to fix some finishing issues and to speed glue/wood filler curing, and the soldering iron to solder the electronics. The heat gun is remarkably quiet and heated up fast, and the wand style makes it really easy to control, as compared to the traditional hair-dryer shape of most heat guns. The soldering iron likewise heated up fast, and the temperature control made fresh and re-soldering really easy. The central unit is about 4x6x8", making storage very easy. Interchangable tips are included for both the soldering iron and heat gun. Obviously you can get a heat gun and soldering iron for less individually, but if you want something that takes up less room and does both jobs well, this does the trick at a very fair price.