Although there is no formal electrical definition for a brownout, the generally accepted definition is a decrease in line voltage for a marked period of time. (Compare to a sag, which is typically caused by a sudden load on the circuit and only lasts for a brief period of time.)
Brownouts may be planned by the electric company in order to forcibly reduce power consumption, particularly during peak demand periods. However, these brownouts typically reduce the overall voltage level by a maximum of five percent (i.e. from 120V to 114V). Most appliances are designed to handle this sort of reduction, so the typical consumer would only see a slight dimming of the lights.
On the other hand, significant brownouts -- where the voltage dips a considerable amount -- can be disruptive, and in some cases damaging. They can also result in extremely odd conditions. Frankly, I have always found brownouts to be a lot eerier than a simple blackout, because everything is on, but not quite. Fluorescent lights that are already on may stay lit, but ones that are off probably won't have the juice to kick back on. Halogen lights may dim, flicker, or shut off entirely, while incandescent lights will simply emit a dull yellow glow. (This is, in fact, how dimmer switches operate.) It feels like being in a real-life Quake II map.
Equipment that relies on a transformer/power rectifier to convert AC into DC acts extremely unpredictably, depending on the design of the power supply and the particular loss of voltage. Some will exhibit no ill effects whatsoever; others will work "partially", as some power rails are generated by the supply while others fail. There may be power enough to light up the pilot LED, but the rest of the system will be dead. In our office here, our server computer acted as if nothing was wrong, while my workstation started flipping out, despite the fact that they were both on the same circuit. Most dangerously, many lights couldn't operate on the low voltage, but the emergency lights still sensed voltage on the mains so they didn't come on.
This brings me to my next point: damage. Equipment that doesn't realize there's insufficient voltage can end up damaging itself as it tries (unsuccessfully) to operate normally. In the case of my workstation, the power supply was barely adequate to supply the low-current rails, until the computer attempted to spin up the hard drives. At this point, the low-current rails would fail, power to the logic circuits would stop, and the spin-up attempt would cease... at which point power would return to the logic circuits, and it would try again, ad infinitum. Result? A hard drive executing a rapid fire parking and unparking of its head; not exactly what you want to expose your equipment to.
So, in the end, what should you do to protect against brownouts? Quite simply, buy a UPS and stick your vital systems on it. Prolonged brownouts can be far more dangerous to sensitive systems than sudden blackouts, since once the blackout is in progress, everything is off. For things not attached to a UPS, shut them off and disconnect them from the power lines. This goes especially for any system with an ATX power supply (almost any system made within the past five years). Brownouts can confuse the hell out of soft power circuits. Finally, be prepared, the same way you would be for a blackout. Keep candles and flashlights (or torches if you're so inclined) ready, and don't attempt to turn on appliances until you're sure full voltage has been restored. Doing this will keep you and your equipment safe.
Addendum: Very brief, localized blackouts are sometimes referred to as "brownouts"; however, the common usage of the term denotes a voltage drop, not a loss of voltage.