Should you find your horn out of order, repair is pretty simple.  A horn is a pretty robust unit, so there is not much that is beyond repair.  Electrically, a horn is merely an electromagnet coil in series with a set of contact points.  As the electromagnet is energized, it pulls a diaphragm inwards.  The diaphragm pushes the contact points open, shutting off the electromagnet and relaxing the diaphragm to repeat the process as the points close again.  The rapid oscillation of the diaphragm creates the sound we all know and love.

Here is a cross section scan, from a 1954 Motor Service's Automotive Encyclopedia, showing the internal components.  The projector, or bugle shaped extension, is shaped differently on the Auto-Lite HA series horn on my CJ-2A:


Six small bolts hold the two halves together.  This view shows the diaphragm held above the rear case half.  A thick steel disc, called the armature, is riveted to the diaphragm and is attracted by the electromagnet.  Note the protruding finger on the armature which will push open the contacts.  The terminals and all wiring are insulated from the case.  While not seen, the other end of the electromagnet coil is wired to the moveable contact arm.  The wire in the foreground is attached to the fixed contact arm.  Note the tip of the striker screw, which is explained further below:


This close-up shows the details of the insulated terminals.  The insulators on the inside have a shoulder so the terminal never touches the oversized hole. Flat phenolic washers are used on the outside of the case.  The terminals from the vehicle wiring will be sandwiched in place by another pair of nuts, not shown.   Note the two adjusting screws.  The striker screw in the middle limits the travel of the armature.  This mainly controls the volume.  (Thanks Sean!)  The other screw adjusts the tension on the breaker points, controlling the frequency and resultant pitch:


Troubleshooting a dead horn is pretty simple.  Check the wiring to the horn.  One terminal is always powered, and the other provides a ground through the button at the steering wheel.  Because one terminal is always powered, disconnect the battery when removing or replacing the horn.  With power removed, check continuity between the two terminals.  With the horn at rest, the contact points should be closed.  The wire in the electromagnet coil is quite heavy, so there should be almost perfect continuity between the two terminals.  A failure of continuity in that heavy wire is quite unlikely, so suspect dirty contact points if you find excessive resistance.  Here a strip of fine sandpaper is folded in half to clean the contact points.  Back out the screw that adjusts the breaker point tension so the spring isn't damaged when the points are gently pushed open for cleaning.  Wash out any grit left on the contacts with compressed air and electronic cleaner solution.  The contact point assembly is riveted together, so disassembly is not advised.  Unless worn out by an ill-tempered taxicab driver, a simple cleaning of the contacts should be sufficient:


After cleaning the points, I measured less than 0.1 ohms between the terminals.  Pay close attention to the nuts which secure the wiring terminals on the outside of the case.  I found the threads corroded inside the steel nuts.  When the vehicle wiring terminals were sandwiched in place, there was poor continuity to the terminals themselves.  After cleaning the terminal threads and installing new brass nuts, continuity was restored.

Before reassembly, check for continuity between each terminal and ground.  Since the wiring is fully insulated from the case, there should be no continuity.  Damaged insulators at the terminals or melted varnish on the electromagnet coil windings are the likely culprits.  Should you find the varnish insulation damaged, rewinding the coil with new magnet wire is not beyond the capabilities of a determined tinkerer.  Match the wire gauge and number of turns and you should be all set.

Here the horn is being reassembled.  On the forward side of the diaphragm there is an item called a ringer disk.  This is what you see when you look in from the open end.  The ringer disc is used to improve the acoustics of the horn:


Adjusting the tone is not hard, but try to pick a day when nobody else is around.  Lock yourself in the garage with the doors closed, to keep from putting your neighbors over the edge.  There is no point in upsetting them so they call the zoning commission who will then inquire about the fifteen Jeeps in various stages of disrepair around the yard.

The horn must be fully assembled and connected to the special shock mount.  The six bolts holding the two halves together must be securely torqued, or you will get a sick buzzing sound.  Don't ask. If you stuff a rag in the horn to keep the sound down, it will throw your adjustments way off.  Don't ask.  The horn draws a lot of current, about 16 amps, so don't try adjusting it on your workbench with a wimpy DC power supply.  Don't ask.  Don't try holding the horn's edge in a vise, as this will also throw off your adjustments.  You may notice a theme here, but don't ask how I know all this.  Suffice to say, the horn must be fully assembled, and properly installed and powered.  The horn is not polarity sensitive, so it doesn't matter which wire is attached to which terminal:


Adjust the contact spring tension first.  The easiest way is to back it out completely until no more tension is felt on the screw.  Turn it in approximately one-half turn beyond when initial contact is felt, and experiment from there for the best sound.  Even the tension on the lock nut will affect the sound, so be patient.  After that, adjust the striker screw in the middle.  A gap of .025"-040" is specified against the moveable armature.  You can carefully thread the striker screw in until you first feel it contact the armature, and then back it out one full turn.  With 32 threads per inch on the screw, one full turn is .031", right in the middle of the desired range.  The striker screw adjustment is easy enough to set by trial and error so don't worry if you have trouble with this measurement.  You may want to go back and fine tune the contact spring tension, as one adjustment will slightly affect the other.  

None of this is difficult at all, but earplugs will help increase your patience.  Due to the battery power present at one of the terminals, be careful not to touch it with the wrench used on the lock nuts.  Initially I found I could not turn in the striker screw far enough.  It is made of brass, and the threads were damaged at the slot for the screwdriver.  Cleaning up the threads required a thread restoring file, as the oddball 1/4"-32 threads are not found in a typical tap and die set.
 
All those 16 amps on their way to ground passing through the steering column give me the creeps.  That is quite a bit of current that can damage the bearings after prolonged use.  In addition, just a smidge of added resistance can degrade the horn performance.  Life must be pretty good if I worry about stuff like this.  Fortunately, adding a horn relay is an easy fix.  A horn relay is standard equipment on many vehicles.

Here is a horn relay schematic, rendered in glorious TooMuchFreeTimeVision(tm).  Originally the battery supply wire went directly to the horn.  The wire from the steering wheel switch is also rerouted to the relay.  Two new wires are needed, between the relay and horn, and from the horn to ground.  Operation of the relay is simple.  Battery power is always present at the center terminal.  When the steering wheel switch is closed, the coil is energized, routing battery power directly to the horn.  Only the coil current travels through the steering wheel switch and bearings.  I measured an insignificant 0.15 amps, which is less than 1 percent of the original configuration:


The relay cover is easy to remove by prying open 6 metal tabs.  There were no markings on the relay.  I applied the arbitrary labels of H, B, and S for the schematic, but you may find other marks:


I painted the relay housing black so it was less conspicuous.  Here the relay is mounted on the fender bracket.  I could not bring myself to trim those two existing wires, so for now I left them long.  The new ground wire from the horn runs directly to the extra large terminal block on the fender.  This non-stock terminal block has four extra terminals on the bottom that are ganged together, creating a reliable ground for the lighting circuits.  If you need to ground directly to the fender, be aware you could have a poor ground path if there is not bare metal-to-metal contact at every connection all the way back to the battery's ground terminal:


Kudos to you if you have read this far.  More details about horns, including how they sound, can be found on Derek Redmond's CJ3B Page.