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I own a 1999 Civic 4dr sedan. Lately I'll be driving along and the power just goes. The speedometer drops to zero and the thermostat drops. The radio stops - it's like all the power is gone. Then, a second later the power seems to be back on but the engine is not running. If I'mk driving fast I just turn the ignition and it restarts. If I'm going slow, I pull over, put the car in park and then restart the engine.

What could be causing the electrical system to shut down like this?

 
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it could be that your alternator is bad. it could be partially charging your battery thus giving you your "shut down". ive had a similar experience with a 99 chevy cavalier 4dr and i recently took my alternator out and didn't ground the negative wire. half way to UTI and the car shut off like you explained, but was not able to just start it back up :/. you should get your battery and alt tested at autozone/pepboys for free if those check out good then you could try and check your ignition switch. a bad ignition switch can cause the car to turn off at random times. had a similar experince with my fathers 78' buick when going uphill or downhill it would just shut off then throw it into neutral and start it back up on a roll haha. if not any of those im at a loss dude.

 
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I am gonna guess you had this fixed, but for those who are maybe having the same issue. Its no doubt the alternator.
Have had a number of them die. In my civic when it pooped itself the entire dash went out but once i was above a certain RPM it would kick back on. Only a few diodes had burned out so it still functioned at high speeds. In your case its likely a failing diode or short. It can sometimes work to pull apart the alt and clean it up or see whether some wiring etc can be fixed but in all honesty its often easier to just replace. You may get it fixed but likely still have an issue and it could cost you a battery.
The only time disassembling a battery and fixing it has worked was on my 85 caddy.

 
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It is much more likely a faulty battery cable connection, either at the battery, or at the other end of one or both of the cables. If the alternator fails, the battery should normally keep the car going for quite some time, until the battery discharges enough. If the battery cables are faulty (corroded), and don't maintain good connection, the alternator will have to work on its own. The alternator relies on the battery for reserve for the field coil, since the field coil is pulsed on and off very rapidly to create an average magnetism so that the right amount of current is produced for the load demand. If the battery is not present as a back-up, the alternator will be powering itself (the output will be powering the field coil). Any sudden collapse in the electrical system (turning on headlights, brake lights, AC, etc.) will cause a collapse in the field coil's magnetism. If it is great enough, the alternator will not be able to recover, and the system will shut down.

Solution: clean both ends of your battery cables, and any other related main wiring that may have corroded connections.

A failed alternator typically does not cause the car to shut down instantaneously. I have had many a failed alternator on my Civic, and at all times I was able to drive the car 25 miles or more.