BE PREPARED! You should keep equipment backups (as well as other essential items) on hand, ready to go should you have a problem. You should also carry a toolbox that has screw and nut drivers, pliers, cutters, jumper wires, plenty of audio adapters, fuses and light bulbs.
For the average system you should have at least one backup cable for every type of cable in your system. The most common system failures can be traced to bad cables. You plug them in and out, you step on them and run over them with your hand truck. Do yourself a big favor and carry backup cables. If you use a wireless mic, keep a wired mic handy as a spare. I find that a Sony Discman can be your best friend when a CD player fails. Toss one in your toolbox (carefully) along with a selection of premixed or compilation CDs just in case your mixer should go down. As a backup, I suggest having a small mixer available.
Now here is where things get a little bit more involved your amplifier. Next to cables, the amplifier is the most common item to fail. Never skimp on a spare amp either; get a professional model. It doesn't have to be big, bad, and brawny, but it should be as reliable as your main amp. Whatever you do, don't try to use a home stereo system. They are not built for the heavy demands of pro sound. A used, professional amplifier, even if it has half the power of your main amp, will be good enough to get you out of a bind. After all, a little music is better than no music.
Speakers are another matter when it comes to backups. Unless you have a large van with room for a spare set of speakers, carrying a second set around is probably not feasible. The best thing to do is to protect your speakers from burning out in the first place by using a compressor/limiter or speaker fuses.
If you have extra room in your car or van, you might want to carry a spare tweeter or midrange, as these are the most common parts of the speaker to burn out and are small enough to put under a car seat. If you work alone and have a problem BE COOL! Don't look at the guests; look immediately at your system. Begin tracing where the problem may be; it might be as simple as you hitting the wrong button! Work as fast as you can to solve the problem. Then when everything is running again, return to the festivities. Quickness is of the utmost importance. Practice this at home with a friend. Have him "cause" a problem on your system and you try to figure it out. Pretty soon you will become familiar with potential problems and, should it occur during a real live performance, you will recognize it and solve the problem quickly.
If you've checked and determined it wasn't user error, you should next check your cables. A typical sign of a bad cable is having a channel suddenly drop out without the amp going into protect. You may also get a hum or buzz in the system suddenly. Do a physical inspection of the cables first. Make sure you have no frayed ends. Then start tracing your way backwards. Start with the speakers; if you have one channel dead or humming, swap the cables. If the problem moves from one speaker to the other, the speaker is good and the problem is further up, or in the speaker wire itself.
Next, swap the cables at the amplifier's output. If the problem shifts between the speakers, then the problem lies further up the chain and is not the wire. Move on to the amp. Make sure it is getting a signal from the mixer. Are the meters moving like they normally should? If so, then you have either a bad channel on the amp, or a bad cable. Follow the same procedure for determining if you have a bad cable to check your speakers. You should shut off your amp every time you change wires to avoid any pops or clicks from surging through your system. If you determine that you have sound coming out of the mixer, and the wires are OK, the problem is in the amp. It is time to break out your spare amp.
If your mixer's outputs are fine, check to see if the problem is isolated to one source (CD player, turntable, etc.). If you have sound coming out on both channels on everything but one unit, then the problem could be in its connecting cable or the unit itself. If so, break out the Sony Discman. If not, then there is a problem with the mixer!
If you are using turntables, sometimes you may get a loose connection in the tonearm or needle. Make sure the headshell is firmly seated into the tonearm and that the contacts are clean. Also, if you are getting a nasty hum out of the system, check the turntable's ground wires.
Should your microphone go bad, and you don't have a spare, you can effectively use your headphone as a mic. It will work fine, although the sound will not be very good. Always keep a spare headphone in your tool kit. Keep an assortment of Adapta-Plugs on hand as well.
This is a pretty straightforward approach should you have a problem with one channel. If both channels are dead or humming, the first place to look is the amp. Is it getting an output from the mixer? You can determine this if the meters are moving and the amp is on, but you get no sound. If so, then the problem could lie in the amplifier. See if the amp is in its "protect" mode.
Many amplifiers have this built-in; it is designed to protect the amplifiers from bad loads or short circuits. You could have a wire that is shorting at the terminals or is frayed. If the two bare wires touch together, that most likely will shut down the amp. The best way to deal with problems on the job is to avoid them in the first place. Don't try to use the same system you use for a 50-person backyard party as you would use for a 300-person high school prom. These are totally different gigs which need totally different systems. If you push your system beyond its limits, you will damage it. Don't shout in your mic, and watch those clip lamps! (Most amplifiers have an overload indicator called "clip." If it lights, lower the volume!) And REMEMBER! Be prepared, and don't panic!